Pontryagin duality (again)
Timothy Murphy
for any locally compact abelian group,
ie how does one show that any continuous homomorphism
G* -> T arises from some g in G
as chi -> chi(g)?
I've looked at a couple of books,
and they both go into Banach algebra.
Is there a simpler or more direct proof?
--
Timothy Murphy
e-mail: gayleard /at/ eircom.net
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
ObiONE
David C. Ullrich
wrote:
>What is the simplest proof that G** = G
>for any locally compact abelian group,
>ie how does one show that any continuous homomorphism
>G* -> T arises from some g in G
>as chi -> chi(g)?
>
>I've looked at a couple of books,
>and they both go into Banach algebra.
Not that it matters, but "go into Banach
algebra" is wrong, like "go into group"
would be wrong. There's no topic or technique
called "Banach algebra", rather _a_ Banach
algebra is a certain sort of object...
>Is there a simpler or more direct proof?
Hmm. Last time this came up various proofs were
too hard because they used Arzela-Ascoli. Now
using Banach algebras makes something too hard.
If there were a well-known proof that used just
high-school algebra people would give it in those
books. Sometimes math is hard.
Hint: When you see something proved using the
theory of Banach algebras that's often because
that proof is much simpler than a proof without
Banach algebras. A classical example:
Say f has period 2pi and never vanishes; let
g = 1/f. Say a_n is the n-th Fourier coefficient
of f and b_n is the n-th Fourier coefficient of g.
If the sum of |a_n| is finite then the sum of |b_n|
is finite.
That's very simple using Banach algebras. How
do you prove it without them?
Q: If I don't know an answer to your question why don't
I just shut up?
A: Because it seems to me you simply have the wrong
attitude towards a few things - not morally wrong,
wrong in the sense that you'd be much better
off without them. When you're wondering
about proving something involving compactness in
function spaces you shouldn't be disappointed to
see Arzela-Ascoli coming up, you should _expect_
it to come up; when you see a proof involving
Banach algebras you shouldn't be disappointed,
you should say "Hooray, a proof by Banach
algebras, this may be simple enough to actually
understand!"
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Timothy Murphy
>>What is the simplest proof that G** = G
>>for any locally compact abelian group,
>>ie how does one show that any continuous homomorphism
>>G* -> T arises from some g in G
>>as chi -> chi(g)?
>>
>>I've looked at a couple of books,
>>and they both go into Banach algebra.
>
> Not that it matters, but "go into Banach
> algebra" is wrong, like "go into group"
> would be wrong. There's no topic or technique
> called "Banach algebra", rather _a_ Banach
> algebra is a certain sort of object...
I might have expressed myself inelegantly,
but I think my meaning was perfectly clear,
and you are being rather pedantic.
I was actually looking at Loomis, "Abstract Harmonic Analysis",
when I made this remark.
This book bases Pontryagin duality on the theory of Banach algebras.
There is nothing wrong with this,
but it was not what I was looking for.
> Hmm. Last time this came up various proofs were
> too hard because they used Arzela-Ascoli. Now
> using Banach algebras makes something too hard.
I didn't say it was too hard.
In fact it is quite straightforward,
but not what I am looking for.
On Ascoli's Theorem, I admit that having looked into the matter
more thoroughly, this is indeed the simplest and most natural way
of dealing with compactness in function spaces,
and so showing G* to be locally compact.
In other words, you were right (on that) and I was wrong.
> If there were a well-known proof that used just
> high-school algebra people would give it in those
> books. Sometimes math is hard.
I'm looking for the most direct proof possible
that G** = G.
It doesn't have to be "simple", just direct,
ie not based on some other theory,
like the theory of Banach algebras.
Maybe there is no such proof; I shall see.
> Hint: When you see something proved using the
> theory of Banach algebras that's often because
> that proof is much simpler than a proof without
> Banach algebras.
There are lots of places in mathematics
where a result follows easily
as part of a general theory,
eg the solution of quartic polynomial equations
follows easily from galois theory,
but that doesn't mean one needs to go into galois theory
in order to solve the problem.
> A classical example:
>
> Say f has period 2pi and never vanishes; let
> g = 1/f. Say a_n is the n-th Fourier coefficient
> of f and b_n is the n-th Fourier coefficient of g.
> If the sum of |a_n| is finite then the sum of |b_n|
> is finite.
>
> That's very simple using Banach algebras. How
> do you prove it without them?
Well, to me that problem looks like a problem in Banach algebra,
so it is not surprising (to me) that this is the way
to solve it.
But the statement that G** = G does not look to me
like a problem in Banach algebra.
[It looks to me like a problem in group representation theory;
and I would prefer a proof not going too far outside that theory,
if there is such a thing.]
> Q: If I don't know an answer to your question why don't
> I just shut up?
I didn't say that, or even think it.
> A: Because it seems to me you simply have the wrong
> attitude towards a few things - not morally wrong,
> wrong in the sense that you'd be much better
> off without them. When you're wondering
> about proving something involving compactness in
> function spaces you shouldn't be disappointed to
> see Arzela-Ascoli coming up, you should _expect_
> it to come up; when you see a proof involving
> Banach algebras you shouldn't be disappointed,
> you should say "Hooray, a proof by Banach
> algebras, this may be simple enough to actually
> understand!"
1) On Ascoli's Theorem, I was wrong.
[In defence, I do think that the 2-page proof
that G* is locally compact,
from which the issue arose,
<www.math.uconn.edu/~kconrad/blurbs/gradnumthy/loccptascoli.pdf>,
could be more clearly expressed.]
2) I understand Loomis' proof that G** = G.
I am looking for alternative proofs.
David C. Ullrich
wrote:
>David C. Ullrich wrote:
>
>>>What is the simplest proof that G** = G
>>>for any locally compact abelian group,
>>>ie how does one show that any continuous homomorphism
>>>G* -> T arises from some g in G
>>>as chi -> chi(g)?
>>>
>>>I've looked at a couple of books,
>>>and they both go into Banach algebra.
>>
>> Not that it matters, but "go into Banach
>> algebra" is wrong, like "go into group"
>> would be wrong. There's no topic or technique
>> called "Banach algebra", rather _a_ Banach
>> algebra is a certain sort of object...
>
>I might have expressed myself inelegantly,
>but I think my meaning was perfectly clear,
>and you are being rather pedantic.
I _said_ "not that it matters". I pointed out
that "using Banach algebra" isn't quite right
assuming that you didn't realize this, and
assuming that you'd want to know that it
wasn't quite right.
It was proved before Banach algebras were invented. (Unless
I'm confusing it with some other traditional application
of Banach algebras - I know one of them has this property.)
>But the statement that G** = G does not look to me
>like a problem in Banach algebra.
Aargh. "in Banach algebra". When you say that it really
_is_ like fingernails scraping on a blackboard. Suppose
you were talking to someone who insisting on talking
about whether or not something was a problem in group...
Anyway, not to say that it _is_ a problem in Banach
algebras, but it seems not at all surprising to me that there
should be a connection, since the characters of G are
the complex homomorphisms of L^1(G).
>[It looks to me like a problem in group representation theory;
(not that it's usually called that for commuttative groups, but)
precisely. One point of view on the point to representation
theory is we learn things about the structure of G by
studying the structure of things like L^2(G). It happens a lot
in general that gizmo(X) tells us things about X. The elements
of L^1(G) are kinda sorts like smoothed-out elements of
G; L^p(G) has the advantage that it's a vector space. So learning
about G* by looking at it in terms of L^1(G) seems like a
perfectly natural thing to do, precisely _because_ we're
talking about group representation theory.
At least that's one way to look at it. The action of G on L^1(G)
is a very natural representation of G...
David C. Ullrich
Timothy Murphy
>>But the statement that G** = G does not look to me
>>like a problem in Banach algebra.
>
> Aargh. "in Banach algebra". When you say that it really
> _is_ like fingernails scraping on a blackboard.
I don't understand your problem with this expression.
Is English your native language?
It seems to me that "in Banach algebra"
is a perfectly acceptable usage for "in the theory of Banach algebras".
Would you object to someone saying "a problem in Lie algebra"?
Or "a problem in algebraic topology"?
Jack Schmidt
> is a perfectly acceptable usage for "in the theory of
> Banach algebras".
>
> Would you object to someone saying "a problem in Lie
> algebra"?
> Or "a problem in algebraic topology"?
I cannot describe exactly the problem, but I agree that
both "in Banach algebra" and "in Lie algebra" sound
awful. However, "in abstract algebra" and such sound
absolutely normal.
My suspicion is that if a mathematical term is also
the title of a course, then it can be used directly
(in my own personal dialect of English), but otherwise
it must be made plural.
Jacobson's book is called "Lie Algebras" not "Lie
Algebra", but there are many books titled "Abstract
Algebra" or "Basic Algebra" (or "Algebraic topology").
I somewhat doubt this is a common mistake, but for
what it is worth:
I remember the first time you wrote to me I was surprised
your English was so perfect, being that you were from
*Iceland*. Of course it didn't take too long before
my eyes regained focus and I could tell the "c" from the
"r" and remember that ".ie" is not in fact Iceland. I
don't remember if your sig at the time mentioned Dublin,
but it very well might have. I tend to stop reading as
soon as it stops being about group theory :) It is
possible someone was just kindly helping a presumed
non-native speaker learn a weird rule of English.
Jack Schmidt
> The elements of L^1(G) are kinda sorts like
> smoothed-out elements of G; L^p(G) has the
> advantage that it's a vector space. So learning
> about G* by looking at it in terms of L^1(G) seems
> like a perfectly natural thing to do, precisely
> _because_ we're talking about group representation
> theory.
>
> At least that's one way to look at it. The action of
> G on L^1(G) is a very natural representation of G...
I am trying to get a feel for how analysts view
representations of groups, and this part struck me
as important to understand.
For a finite group, all of the L^p(G) are basically
the same, right? The representation is called the
regular representation, C[G].
I wasn't sure if in your message you meant to be
contrasting L^1(G) with L^p(G), or rather just with
G itself. By G*, we mean C^0(G;C), the space of
continuous functions from G? L^p(G) then being
various completions of G*?
If G were something like GL(n,C) [ n=1 if we need G
to be abelian ]. Roughly how does L^p(G) change with
p?
If it helps here is approximately what I remember
from 4 years of training to be an analyst:
I think with n=1 L^1 is the simple Fourier series
that works extremely obviously when restricted to
continuous functions, L^2 is the better Fourier series
with an inner product, but I'm not sure what happens
when n is not 1 or p is neither 1 nor 2. Indeed, going
from p=1 to p=2 was just another random time we needed
to complete a space to have solutions to some PDE, so
I wouldn't say I understand p=1 or p=2 either. The
motivating problem was strictly with p=1, and the rules
for working with them were strictly p=2. Also, there is
no guarantee I remember the difference between G=S^1,
G=R^1, and G=R^n.
Timothy Murphy
> For a finite group, all of the L^p(G) are basically
> the same, right? The representation is called the
> regular representation, C[G].
To my mind, L^2(G) has a special place in representation theory,
even if G is finite,
since this makes explicit the invariant quadratic or hermitian form.
> I wasn't sure if in your message you meant to be
> contrasting L^1(G) with L^p(G), or rather just with
> G itself. By G*, we mean C^0(G;C), the space of
> continuous functions from G? L^p(G) then being
> various completions of G*?
Well, G* is the group of continuous homomorphisms chi:G->T,
ie |chi(g)| = 1.
Jack Schmidt
>
> > For a finite group, all of the L^p(G) are basically
> > the same, right? The representation is called the
> > regular representation, C[G].
>
> To my mind, L^2(G) has a special place in
> representation theory,
> even if G is finite,
> since this makes explicit the invariant quadratic or
> hermitian form.
I usually avoid the characteristic zero representations
if possible, and just let the inner product of characters
do all my orthogonality stuff.
However, I am getting interested in the modular version
of this. I don't it is phrased in terms of L^2 at all,
but rather in terms of unitary representations, like
Hom( G, U(n,C) ) gets replaced with Hom( G, U(n,q) ),
or Hom( G, X(n,q) ) for some classical group X(n,q).
> > I wasn't sure if in your message you meant to be
> > contrasting L^1(G) with L^p(G), or rather just with
> > G itself. By G*, we mean C^0(G;C), the space of
> > continuous functions from G? L^p(G) then being
> > various completions of G*?
>
> Well, G* is the group of continuous homomorphisms
> chi:G->T, ie |chi(g)| = 1.
Oh that's pretty different, but it makes sense. I was
thinking of a finite abelian group G, and then the space
of class functions on G (all of which are continuous,
since G is discrete). Since G is abelian, this was just
yet another way of writing down C[G]. I was sort of
hoping the L^p(G) when G was infinite would be
completions.
I think though this cannot be what DCU intended as
I am apparently talking about the C-vector space
generated by G*. That is pretty twisted if we are
thinking of G* as a locally compact abelian group
(which is what the subject of this thread states,
right?).
David C. Ullrich
wrote:
>David C. Ullrich wrote:
>
>>>But the statement that G** = G does not look to me
>>>like a problem in Banach algebra.
>>
>> Aargh. "in Banach algebra". When you say that it really
>> _is_ like fingernails scraping on a blackboard.
>
>I don't understand your problem with this expression.
>Is English your native language?
I should be asking you that question.
>It seems to me that "in Banach algebra"
>is a perfectly acceptable usage for "in the theory of Banach algebras".
I understand that it seems that way to you. That's why
I pointed out that it's not so, assuming that you'd want
to know.
>Would you object to someone saying "a problem in Lie algebra"?
Yes. At least I'm pretty sure this is exactly as obhectionable
as "in Banach algebra", although I'm not certain.
>Or "a problem in algebraic topology"?
No. The difference is that "algebraic topology" is the standard
name for a certain branch of mathematics. "Banach algebra"
is _not_ the standard name for a certain branch of mathematics
(and I'm pretty sure but not quite as certain that "Lie algebra"
is also not such a name).
Do _you_ see a problem with "a problem in group"? If
no, there's no point in continuing this. If yes: the problem
with "a problem in Banach algebra" (and what I'm pretty
sure is the problem with "a problem in Lie algebra") is
exactly the same. If you do see a problem with "a
problem in group" but don't feel that "a problem in
Banach algebra" is equally bad that's just because you're
not familiar with what is and what is not standard usage.
There's simply no such thing as "Banach algebra",
just as there's no such thing as "group".
David C. Ullrich
<Jack.Schmi...@gmail.com> wrote:
>[ G a locally compact abelian group, I think ]
>> The elements of L^1(G) are kinda sorts like
>> smoothed-out elements of G; L^p(G) has the
>> advantage that it's a vector space. So learning
>> about G* by looking at it in terms of L^1(G) seems
>> like a perfectly natural thing to do, precisely
>> _because_ we're talking about group representation
>> theory.
>>
>> At least that's one way to look at it. The action of
>> G on L^1(G) is a very natural representation of G...
>
>I am trying to get a feel for how analysts view
>representations of groups, and this part struck me
>as important to understand.
>
>For a finite group, all of the L^p(G) are basically
>the same, right?
Yes or no, depending on what sort of sameness
you mean. They're the same as vector spaces.
There'r isomophic as normed vector spaces,
although not isometric as normed vector
spaces. And of course L^2 is special, being
a Hilbert space, which the others are not.
>The representation is called the
>regular representation, C[G].
>
>I wasn't sure if in your message you meant to be
>contrasting L^1(G) with L^p(G), or rather just with
>G itself.
The point was sort of that I know the OP is going to
agree that L^2(G) has a lot to do with studying G.
In fact other p can be relevant, and in particular
taking p = 1 explains why getting at G* via
Banach algebraS is perfectly natural.
>By G*, we mean C^0(G;C), the space of
>continuous functions from G? L^p(G) then being
>various completions of G*?
No, we're assuming G is locally compact abelian,
and G* is the dual group: the group of continuous
homorphisms of G into the unit circle in the complex
plane.
>If G were something like GL(n,C) [ n=1 if we need G
>to be abelian ]. Roughly how does L^p(G) change with
>p?
>
>If it helps here is approximately what I remember
>from 4 years of training to be an analyst:
>
>I think with n=1 L^1 is the simple Fourier series
>that works extremely obviously when restricted to
>continuous functions, L^2 is the better Fourier series
>with an inner product, but I'm not sure what happens
>when n is not 1 or p is neither 1 nor 2. Indeed, going
>from p=1 to p=2 was just another random time we needed
>to complete a space to have solutions to some PDE, so
>I wouldn't say I understand p=1 or p=2 either. The
>motivating problem was strictly with p=1, and the rules
>for working with them were strictly p=2. Also, there is
>no guarantee I remember the difference between G=S^1,
>G=R^1, and G=R^n.
David C. Ullrich
David C. Ullrich
In the case of a locally compact abelian group
the _irreducible_ unitary representations are
all one-dimensional, ie representations in
U(1,C). But U(1,C) is naturally isomorphic
to the unit circle in the plane, and there you are.
> I was
>thinking of a finite abelian group G, and then the space
>of class functions on G (all of which are continuous,
>since G is discrete). Since G is abelian, this was just
>yet another way of writing down C[G]. I was sort of
>hoping the L^p(G) when G was infinite would be
>completions.
>
>I think though this cannot be what DCU intended as
>I am apparently talking about the C-vector space
>generated by G*. That is pretty twisted if we are
>thinking of G* as a locally compact abelian group
>(which is what the subject of this thread states,
>right?).
We can also think of the elements of G* as functions
from G to C. In the compact case they are in L^p(G)
and yes, L^p(G) is the completion of their span,
at least for 1 <= p < infinity; for p = infinity the
completion of the span of G* is the space
of continuous functions.
In the non-compact case the elements of G* are
not elements of L^p for 1 <= p < infinity.
They are in L^infinity; the completion of their
span in L^infinity is the space of "almost periodic"
functions.
Timothy Murphy
>>>>But the statement that G** = G does not look to me
>>>>like a problem in Banach algebra.
>>>
>>> Aargh. "in Banach algebra". When you say that it really
>>> _is_ like fingernails scraping on a blackboard.
>>
>>I don't understand your problem with this expression.
>>Is English your native language?
>
> I should be asking you that question.
English is my native language.
But perhaps you should have answered my question first.
>>It seems to me that "in Banach algebra"
>>is a perfectly acceptable usage for "in the theory of Banach algebras".
>
> I understand that it seems that way to you. That's why
> I pointed out that it's not so, assuming that you'd want
> to know.
I don't believe in "proof by assertion".
Do you have any qualifications in this subject (English usage)
to enable you to pontificate in this way?
>
>>Would you object to someone saying "a problem in Lie algebra"?
>
> Yes. At least I'm pretty sure this is exactly as obhectionable
> as "in Banach algebra", although I'm not certain.
>
>>Or "a problem in algebraic topology"?
>
> No. The difference is that "algebraic topology" is the standard
> name for a certain branch of mathematics. "Banach algebra"
> is _not_ the standard name for a certain branch of mathematics
> (and I'm pretty sure but not quite as certain that "Lie algebra"
> is also not such a name).
What name do you give to the study of Lie algebras?
> Do _you_ see a problem with "a problem in group"? If
> no, there's no point in continuing this.
If you don't see the difference, there is indeed not much point
in continuing the discussion.
"Algebra" is a branch of mathematics,
and "Lie" is used here as an adjective describing a sub-species
of algebra.
You may use "group" to describe a branch of mathematics,
but most people do not.
Much more to the point: if you have no contribution to make
in answer to a query - and you have said nothing so far on this topic
that has struck me as the slightest bit helpful -
it is simply an exercise in uncivilized rudeness
to criticize the way the question was posed.
David Bernier
For the word "algebra", there is the usage as
a branch of mathematics, and another usage as
a type of algebraic structure; in the
second usage, an algebra is a type of ring
which is also a vector space over a field k, with
accompanying axioms. Also, topology can refer to
a branch of mathematics or to a pair (X, O)
with X a set and O a collection of subsets of X such
that (list of axioms) is satisfied. We can imagine
the expression "Banach topologies" but it has no
accepted meaning, so it's probably meaningless.
So in summary, when I think of "Banach algebra",
I'm thinking of a type of algebra
(algebra <==> ring & vector space).
David Bernier
David C. Ullrich
wrote:
>David C. Ullrich wrote:
>
>>>>>But the statement that G** = G does not look to me
>>>>>like a problem in Banach algebra.
>>>>
>>>> Aargh. "in Banach algebra". When you say that it really
>>>> _is_ like fingernails scraping on a blackboard.
>>>
>>>I don't understand your problem with this expression.
>>>Is English your native language?
>>
>> I should be asking you that question.
>
>English is my native language.
>But perhaps you should have answered my question first.
>
>>>It seems to me that "in Banach algebra"
>>>is a perfectly acceptable usage for "in the theory of Banach algebras".
>>
>> I understand that it seems that way to you. That's why
>> I pointed out that it's not so, assuming that you'd want
>> to know.
>
>I don't believe in "proof by assertion".
>Do you have any qualifications in this subject (English usage)
>to enable you to pontificate in this way?
I've read about Banach algebras for many years. I've
never seen the phrase "Banach algebra" used in the
sense in which you're using it here except in this thread.
I just entered "Banach algebra" into google. On the first
page of ten hits I saw nine uses of the phrase in its
standard sense and none in the sense in which you're
using it.
Find one place where it _is_ used in the sense in which
you're using it.
>>>Would you object to someone saying "a problem in Lie algebra"?
>>
>> Yes. At least I'm pretty sure this is exactly as obhectionable
>> as "in Banach algebra", although I'm not certain.
>>
>>>Or "a problem in algebraic topology"?
>>
>> No. The difference is that "algebraic topology" is the standard
>> name for a certain branch of mathematics. "Banach algebra"
>> is _not_ the standard name for a certain branch of mathematics
>> (and I'm pretty sure but not quite as certain that "Lie algebra"
>> is also not such a name).
>
>What name do you give to the study of Lie algebras?
"The study of Lie algebras".
>> Do _you_ see a problem with "a problem in group"? If
>> no, there's no point in continuing this.
>
>If you don't see the difference, there is indeed not much point
>in continuing the discussion.
>
>"Algebra" is [i] a branch of mathematics,
And it's also [ii] the name of a certain sort of algebraic structure.
>and "Lie" is used here as an adjective describing a sub-species
>of algebra.
The standard usage is as an adjective describing a sub-species
of "algebra" in sense [ii]. This thread is the first time I've
seen it used to modify "algebra" in sense [i].
You should note again that I'm not nearly as certain
here about "Lie algebra" as I am about "Banach algebra".
>You may use "group" to describe a branch of mathematics,
>but most people do not.
"Most" people do not? Name one person who does.
>Much more to the point: if you have no contribution to make
>in answer to a query - and you have said nothing so far on this topic
>that has struck me as the slightest bit helpful -
My comments on your comments on the use of the A-A theorem
didn't appear to strike you as helpful either, at first - now you're
agreeing that I was simply right about that. Your opinion about
my comments on this topic may change as well when you calm
down.
Read what I had to say when I saw the phrase "representation
theory" in one of your posts here.
>it is simply an exercise in uncivilized rudeness
>to criticize the way the question was posed.
Calling my comments "rude" is rude. I began
with "not that it matters". Your usage _is_
extremely non-standard here - one would think
that a rational person would want to be informed
when he's using the language incorrectly.
Which would you prefer: I let you know, gently at
first, that the phrase is simply not used that way,
or that you sound illiterate some day when you're
giving a talk or writing a paper?
"Uncivilized rudeness" - I like that. What I said
was this:
'Not that it matters, but "go into Banach
algebra" is wrong, like "go into group"
would be wrong. There's no topic or technique
called "Banach algebra", rather _a_ Banach
algebra is a certain sort of object...'
What I didn't say was this:
'"go into Banach algebra"? Wow, is English
your second language or what? Learn to write,
guy.'
See the difference?
Timothy Murphy
>>You may use "group" to describe a branch of mathematics,
>>but most people do not.
>
> "Most" people do not? Name one person who does.
You, I assume, since you introduced the phrase "in group".
>>Much more to the point: if you have no contribution to make
>>in answer to a query - and you have said nothing so far on this topic
>>that has struck me as the slightest bit helpful -
>
> My comments on your comments on the use of the A-A theorem
> didn't appear to strike you as helpful either, at first
They weren't helpful.
They were correct.
There is a difference.
> Read what I had to say when I saw the phrase "representation
> theory" in one of your posts here.
I'm afraid I didn't see anything helpful.
Remind me, I must have missed it.
>>it is simply an exercise in uncivilized rudeness
>>to criticize the way the question was posed.
>
> Calling my comments "rude" is rude. I began
> with "not that it matters".
You mean, it would make it alright if I said,
"Not that it matters, but you are a pedantic prig"?
Regarding the question at issue,
whether one can speak of Banach algebra or Lie algebra
as describing a certain branch of algebra,
may I recommend Rule 3 in George Orwell's
'Politics and the English Language':
"If it is possible to cut a word out, always cut it out."
As a matter of interest, how would you re-write
"The Campbell-Hausdorff-Baker theorem
is a result in Lie algebra".
Michael Press
Timothy Murphy <t...@maths.tcd.ie> wrote:
> David C. Ullrich wrote:
>
> >>You may use "group" to describe a branch of mathematics,
> >>but most people do not.
> >
> > "Most" people do not? Name one person who does.
>
> You, I assume, since you introduced the phrase "in group".
>
> >>Much more to the point: if you have no contribution to make
> >>in answer to a query - and you have said nothing so far on this topic
> >>that has struck me as the slightest bit helpful -
> >
> > My comments on your comments on the use of the A-A theorem
> > didn't appear to strike you as helpful either, at first
>
> They weren't helpful.
> They were correct.
> There is a difference.
If they are correct, and you discovered that
they are entirely correct, then they are helpful.
By most definitions of helpful.
You may feel pique, or something stronger,
but you know that you will get over it.
--
Michael Press
Michael Press
> As a matter of interest, how would you re-write
> "The Campbell-Hausdorff-Baker theorem
> is a result in Lie algebra".
"The Campbell-Hausdorff-Baker theorem
is a result on Lie algebras".
--
Michael Press
Timothy Murphy
>> > My comments on your comments on the use of the A-A theorem
>> > didn't appear to strike you as helpful either, at first
>>
>> They weren't helpful.
>> They were correct.
>> There is a difference.
>
> If they are correct, and you discovered that
> they are entirely correct, then they are helpful.
> By most definitions of helpful.
The comment was correct.
It was not helpful because I already had the information.
Does it matter?
Timothy Murphy
I believe my expression is (slightly) more accurate.
The theorem is a deduction from the axioms for a Lie algebra.
and does therefore hold for all Lie algebras.
But there might (in theory) be other cases where the axioms,
and therefore the result, holds,
eg in some category other than that of finite dimensional
vector spaces.
It is, in other words, a result in that branch of algebra
that deals with the consequences of the Jacobi identity,
and it is reasonable and rational to use the term
"Lie algebra" for that branch of algebra.
It is the difference between saying that a theorem
is a result in topology, and saying that it is
a theorem on topological spaces.
A minuscule difference, but a difference none the less.
David C. Ullrich
wrote:
>David C. Ullrich wrote:
>
>>>You may use "group" to describe a branch of mathematics,
>>>but most people do not.
>>
>> "Most" people do not? Name one person who does.
>
>You, I assume, since you introduced the phrase "in group".
I never _used_ that phrase, I _mentioned_ it as an example
to illustrate what's wrong with "in Banach algebra:.
And you understand that very well.
>>>Much more to the point: if you have no contribution to make
>>>in answer to a query - and you have said nothing so far on this topic
>>>that has struck me as the slightest bit helpful -
>>
>> My comments on your comments on the use of the A-A theorem
>> didn't appear to strike you as helpful either, at first
>
>They weren't helpful.
>They were correct.
>There is a difference.
>
>> Read what I had to say when I saw the phrase "representation
>> theory" in one of your posts here.
>
>I'm afraid I didn't see anything helpful.
>Remind me, I must have missed it.
>
>>>it is simply an exercise in uncivilized rudeness
>>>to criticize the way the question was posed.
>>
>> Calling my comments "rude" is rude. I began
>> with "not that it matters".
>
>You mean, it would make it alright if I said,
>"Not that it matters, but you are a pedantic prig"?
Another hint: what you're accomplishing here is
something that I doubt you _want_ to accomplish.
>Regarding the question at issue,
>whether one can speak of Banach algebra or Lie algebra
>as describing a certain branch of algebra,
>may I recommend Rule 3 in George Orwell's
>'Politics and the English Language':
>"If it is possible to cut a word out, always cut it out."
>
>As a matter of interest, how would you re-write
>"The Campbell-Hausdorff-Baker theorem
>is a result in Lie algebra".
Why do you ask? Has someone with a reasonable
command of English actually said that?
And where's the corresponding example with
"Banach algebra"?
David C. Ullrich
wrote:
>Michael Press wrote:
>
>> In article <gcb2oh$nk2$1...@registered.motzarella.org>,
>> Timothy Murphy <t...@maths.tcd.ie> wrote:
>>
>>> As a matter of interest, how would you re-write
>>> "The Campbell-Hausdorff-Baker theorem
>>> is a result in Lie algebra".
>>
>> "The Campbell-Hausdorff-Baker theorem
>> is a result on Lie algebras".
>
>I believe my expression is (slightly) more accurate.
>The theorem is a deduction from the axioms for a Lie algebra.
>and does therefore hold for all Lie algebras.
>But there might (in theory) be other cases where the axioms,
>and therefore the result, holds,
>eg in some category other than that of finite dimensional
>vector spaces.
>
>It is, in other words, a result in that branch of algebra
>that deals with the consequences of the Jacobi identity,
>and it is reasonable and rational to use the term
>"Lie algebra" for that branch of algebra.
If you really think that what's reasonable and accurate
has some bearing on what is or is not accepted usage
that would explain a lot.
You're actually saying that it's "reasonable and accurate"
to use the term "Lie algebra" to mean something _other_
than the theory of Lie algebras. That's remarkable.
>It is the difference between saying that a theorem
>is a result in topology, and saying that it is
>a theorem on topological spaces.
>
>A minuscule difference, but a difference none the less.
David C. Ullrich
Phil Carmody
Phil
--
The fact that a believer is happier than a sceptic is no more to the
point than the fact that a drunken man is happier than a sober one.
The happiness of credulity is a cheap and dangerous quality.
-- George Bernard Shaw (1856-1950), Preface to Androcles and the Lion
David C. Ullrich
<thefatphi...@yahoo.co.uk> wrote:
>David C. Ullrich <dull...@sprynet.com> writes:
>> On Sun, 05 Oct 2008 19:58:24 +0100, Timothy Murphy <t...@maths.tcd.ie> wrote:
>>>As a matter of interest, how would you re-write
>>>"The Campbell-Hausdorff-Baker theorem
>>>is a result in Lie algebra".
>>
>> Why do you ask? Has someone with a reasonable
>> command of English actually said that?
>>
>> And where's the corresponding example with
>> "Banach algebra"?
>
>google
Thanks. I tried that, didn't find any. What example
do you find on google?
>Phil
Phil Carmody
> On Mon, 06 Oct 2008 17:02:08 +0300, Phil Carmody
> <thefatphi...@yahoo.co.uk> wrote:
>
>>David C. Ullrich <dull...@sprynet.com> writes:
>>> On Sun, 05 Oct 2008 19:58:24 +0100, Timothy Murphy <t...@maths.tcd.ie> wrote:
>>>>As a matter of interest, how would you re-write
>>>>"The Campbell-Hausdorff-Baker theorem
>>>>is a result in Lie algebra".
>>>
>>> Why do you ask? Has someone with a reasonable
>>> command of English actually said that?
>>>
>>> And where's the corresponding example with
>>> "Banach algebra"?
>>
>
> Thanks. I tried that, didn't find any. What example
> do you find on google?
You do actually know how to use google? You type a phrase in
the search box, and click 'Search'?
Or did you simply not consider that the construction
"in Banach Algebra" would be a sensible thing to search
for? In particular, given that that was the precise phrase
to which you were objecting. Were you searching for
"The Campbell-Hausdorff-Baker theorem is a result in
Banach Algebra", perhaps? If so, you got, and are now
getting, what you deserve.
If you did that sensible search, rather than whatever
you did search for, you'll see the top hit is the usage
you assert isn't used. Then if you waded through the
hits that are simply archives of this thread (which is
about half of the hits now), you'll see that about one
hit in ten is using 'Banach Algebra' to mean the field
of Banach Algebra theory, which was clearly what was
meant by Timothy.
David C. Ullrich
<thefatphi...@yahoo.co.uk> wrote:
>David C. Ullrich <dull...@sprynet.com> writes:
>> On Mon, 06 Oct 2008 17:02:08 +0300, Phil Carmody
>> <thefatphi...@yahoo.co.uk> wrote:
>>
>>>David C. Ullrich <dull...@sprynet.com> writes:
>>>> On Sun, 05 Oct 2008 19:58:24 +0100, Timothy Murphy <t...@maths.tcd.ie> wrote:
>>>>>As a matter of interest, how would you re-write
>>>>>"The Campbell-Hausdorff-Baker theorem
>>>>>is a result in Lie algebra".
>>>>
>>>> Why do you ask? Has someone with a reasonable
>>>> command of English actually said that?
>>>>
>>>> And where's the corresponding example with
>>>> "Banach algebra"?
>>>
>>>google
>>
>> Thanks. I tried that, didn't find any. What example
>> do you find on google?
>
>You do actually know how to use google?
Evidently not.
>You type a phrase in
>the search box, and click 'Search'?
>
>Or did you simply not consider that the construction
>"in Banach Algebra" would be a sensible thing to search
>for? In particular, given that that was the precise phrase
>to which you were objecting. Were you searching for
>"The Campbell-Hausdorff-Baker theorem is a result in
>Banach Algebra", perhaps? If so, you got, and are now
>getting, what you deserve.
>
>If you did that sensible search, rather than whatever
>you did search for, you'll see the top hit is the usage
>you assert isn't used. Then if you waded through the
>hits that are simply archives of this thread (which is
>about half of the hits now), you'll see that about one
>hit in ten is using 'Banach Algebra' to mean the field
>of Banach Algebra theory, which was clearly what was
>meant by Timothy.
Let's see. I never asserted that the phrase was never
used, just that it's wrong. It's not hard to find lots
of bad English on the internet.
"Bamach Algebra" gets 88,000 hits, while
"in Banach Algebra" gets 718. I was about to point
out that the fact that that _phrase_ appears doesn't
mean the author is actually using the usage in question,
for example "in Banach Algebra theory" is precisely
the usage I'm claiming is correct, even though it
includes the phrase "in Banach Algebra", but
you've already acknowledged that.
I'm not sure whether you're saying 10% or 5% of
the hits actually involve the disputed usage.
Assuming your estimate is correct, we have
70 or 35 hits using the disputed usage, compared
to 88,000 hits on just "Banach Algebra". Even
if there are that many actual instances of the
disputed usage, that many proves nothing.
Ok, let's look. The first hit points to
http://www.mast.queensu.ca/~mingo/seminar/winter_2005.html
where we see that the "in Banach Algebra" on the hits
page is actually part of "in Banach Algebra Situations".
Not the usage we're talking about.
The second hit gives me "Content Not Found".
(The snippet we see on the hits page does indeed appear
to be an instance of the disputed usage. See hit 7 below
for an explanation of why I consider this to be no
evidence at all unless I can see the actual paper.)
Third hit: "in Banach algebra theory". Nope.
Fourth hit: This thread.
Fifth hit: The "in Banach algebra" we see on the hits
page turns out to be "in Banach algebra A" in the
actual paper. Nope.
Sixth: "in Banach algebra homology". Nope.
Seventh: That appears to be an actual instance of the
usage I'm objecting to. Now look at the whole sentence
on the hits page:
"Since 2 is not torsion in Banach algebra, even-though is identical on
R and = 1, [...]"
Not English as we know it, doesn't really count.
Eight: "in Banach algebra extensions" Nope.
Ninth: "in Banach algebra language" Nope.
Tenth: This thread.
I'm on a dialup connection - I'm not going to try to look at
the other 700 hits. If there are in fact 35 or 70 actual
examples out there, _where_ the writer is using correct
English otherwise, such a small number is still not going
to prove that it's an accepted usage, if anything the
fact that the number's so small would be evidence to
the contrary - if "in Banach Algebra" were correct
we wouldn't see people writing "iu Banach Algebra
theory", as we _do_ above (have you ever seen
someone say "in abstract algebra theory"?)
David C. Ullrich
I posted a reply to what's below yesterday. Anything posted in
about the last day has become invisible to me - if you had a
reply to my reply please re-post it.
On Tue, 07 Oct 2008 15:23:03 +0300, Phil Carmody
<thefatphi...@yahoo.co.uk> wrote:
David C. Ullrich
Phil Carmody
> Never mind the details of the PC explosion and consequences...
>
> I posted a reply to what's below yesterday. Anything posted in
> about the last day has become invisible to me - if you had a
> reply to my reply please re-post it.
From what you posted, I'm reminded that Google returns different
results to different people, depending on where they reside, so,
as, such, comparisons of what it returns are futile.
David C. Ullrich
<thefatphi...@yahoo.co.uk> wrote:
>David C. Ullrich <dull...@sprynet.com> writes:
>> Never mind the details of the PC explosion and consequences...
>>
>> I posted a reply to what's below yesterday. Anything posted in
>> about the last day has become invisible to me - if you had a
>> reply to my reply please re-post it.
>
>From what you posted, I'm reminded that Google returns different
>results to different people, depending on where they reside, so,
>as, such, comparisons of what it returns are futile.
That's very convenient. I say something, you claim that it's
contradicted by google results, then when it's not contradicted
by my google results it turns out that google doesn't count.
Instead of just making claims about what you saw on google
you _could_ simply _give_ us the urls that google gave
_you_ that contradict the "claim" that the disputed usage
is in fact incorrect. Then we could see for ourselves whether
they're actual instances of that usage or non-instances that
nonetheless happen to include the phrase "in Banach
algebra" - we could also see for ourselves whether the
rest of the English in the example was reasonably
standard.
Phil Carmody
> On Fri, 10 Oct 2008 14:53:47 +0300, Phil Carmody
> <thefatphi...@yahoo.co.uk> wrote:
>
>>David C. Ullrich <dull...@sprynet.com> writes:
>>> Never mind the details of the PC explosion and consequences...
>>>
>>> I posted a reply to what's below yesterday. Anything posted in
>>> about the last day has become invisible to me - if you had a
>>> reply to my reply please re-post it.
>>
>>From what you posted, I'm reminded that Google returns different
>>results to different people, depending on where they reside, so,
>>as, such, comparisons of what it returns are futile.
>
> That's very convenient. I say something, you claim that it's
> contradicted by google results, then when it's not contradicted
> by my google results it turns out that google doesn't count.
It's convenient for whom that I state that my own prior
argument has little weight? It's most *inconvenient* for
me, I can assure you.
> Instead of just making claims about what you saw on google
> you _could_ simply _give_ us the urls that google gave
> _you_ that contradict the "claim" that the disputed usage
> is in fact incorrect. Then we could see for ourselves whether
> they're actual instances of that usage or non-instances that
> nonetheless happen to include the phrase "in Banach
> algebra" - we could also see for ourselves whether the
> rest of the English in the example was reasonably
> standard.
Because of this thread, they're a moving target. And what's
this "standard" you're referring to? Got a URL for that?
From the top 10 I'm given:
IngentaConnect Solutions of some functional-integral equations in ...
Using the technique of measures of noncompactness in Banach algebra, we prove art existence theorem for a functional-integral equation. ...
www.ingentaconnect.com/content/els/08957177/2003/00000038/00000003/art90084 - Similar pages
by J Banas - 2003 - Cited by 4 - Related articles - All 3 versions
Home page of Prof. S.H. Kulkarni
G. Ramesh, APPROXIMATION METHODS FOR SOLVING OPERATOR EQUATIONS INVOLVING UNBOUNDED OPERATORS, 2008. D. Sukumar, RANSFORD SPECTRUM IN BANACH ALGEBRA, 2007 . ...
mat.iitm.ac.in/~shk/ - 10k - Cached - Similar pages
Page 3 has:
NEAR-ALGEBRA和BANACH代数上的一个特征值定理Eigenvalue Theorem on ...
- [ Translate this page ]
Some functional equations in Banach algebra and an application 《Proc Amer Soc》 Vukman J 1987 / / 100 P 133-136. - On derivations in prime rings and Banach ...
scholar.ilib.cn/Abstract.aspx?A=dzkjdxxb200506034 - Similar pages
by 杨汉生 - 2005 - Related articles
Page 4 has one dupe and two new ones:
Solutions of some functional-integral equations in Banach algebra
- [ Translate this page ]
Using the technique of measures of noncompactness in Banach algebra we prove an existence theorem for a functional-integral equation. ...
cat.inist.fr/?aModele=afficheN&cpsidt=15110313 - Similar pages
by J BANAS - 2003 - Cited by 4 - Related articles - All 3 versions
[DOC]
Program for the M.Sc. Degree in Mathematics
File Format: Microsoft Word - View as HTML
Banach Algebras, Spectral theory in Banach algebra. Commutative Banach Algebras. Gelfand Mapping. Spectral theorem for normal operators. ...
faculty.ksu.edu.sa/58902/Pictures%20Library/MSC899%5B1%5D.doc - Similar pages
by A Channel - All 2 versions
Department of Mathematics & Statistics
G. Liu M.Sc. 1992 The holder inequality and its applications in Banach algebra and geometric programming. (Dr. C.-L. Wang). B. Ding Ph.D. 1992 Structure and ...
www.math.uregina.ca/graduate/theses.html - 14k - Cached - Similar pages
As I said - about 1 in 10 is using the more compact form. It may
not be the traditional name, but the shorter name loses precisely
zero information, introduces no ambiguity, and is shorter. What's
not to like about it? And if it's being adopted by non-native
English speakers, which seems to be the case, that's probably a
good indication that it's a more obvious name.
David C. Ullrich
<thefatphi...@yahoo.co.uk> wrote:
That's a silly question. The "reference" for what is and
what is not standard mathematical English is the body
of mathematics written in English.
>From the top 10 I'm given:
>
>IngentaConnect Solutions of some functional-integral equations in ...
>Using the technique of measures of noncompactness in Banach algebra, we prove art existence theorem for a functional-integral equation. ...
>www.ingentaconnect.com/content/els/08957177/2003/00000038/00000003/art90084 - Similar pages
>by J Banas - 2003 - Cited by 4 - Related articles - All 3 versions
>
>Home page of Prof. S.H. Kulkarni
>G. Ramesh, APPROXIMATION METHODS FOR SOLVING OPERATOR EQUATIONS INVOLVING UNBOUNDED OPERATORS, 2008. D. Sukumar, RANSFORD SPECTRUM IN BANACH ALGEBRA, 2007 . ...
>mat.iitm.ac.in/~shk/ - 10k - Cached - Similar pages
>
>
>Page 3 has:
>
>NEAR-ALGEBRA?BANACH???????????Eigenvalue Theorem on ...
> - [ Translate this page ]
>Some functional equations in Banach algebra and an application ?Proc Amer Soc? Vukman J 1987 / / 100 P 133-136. - On derivations in prime rings and Banach ...
>scholar.ilib.cn/Abstract.aspx?A=dzkjdxxb200506034 - Similar pages
>by ??? - 2005 - Related articles
>
>Page 4 has one dupe and two new ones:
>
>Solutions of some functional-integral equations in Banach algebra
> - [ Translate this page ]
>Using the technique of measures of noncompactness in Banach algebra we prove an existence theorem for a functional-integral equation. ...
>cat.inist.fr/?aModele=afficheN&cpsidt=15110313 - Similar pages
>by J BANAS - 2003 - Cited by 4 - Related articles - All 3 versions
>
>[DOC]
>Program for the M.Sc. Degree in Mathematics
>File Format: Microsoft Word - View as HTML
>Banach Algebras, Spectral theory in Banach algebra. Commutative Banach Algebras. Gelfand Mapping. Spectral theorem for normal operators. ...
>faculty.ksu.edu.sa/58902/Pictures%20Library/MSC899%5B1%5D.doc - Similar pages
>by A Channel - All 2 versions
>
>Department of Mathematics & Statistics
>G. Liu M.Sc. 1992 The holder inequality and its applications in Banach algebra and geometric programming. (Dr. C.-L. Wang). B. Ding Ph.D. 1992 Structure and ...
>www.math.uregina.ca/graduate/theses.html - 14k - Cached - Similar pages
>
>
>As I said - about 1 in 10 is using the more compact form. It may
>not be the traditional name, but the shorter name loses precisely
>zero information, introduces no ambiguity, and is shorter.
None of which has any relevance to anything I've said about it.
My claim is simply that it's not standard usage.
>What's
>not to like about it? And if it's being adopted by non-native
>English speakers, which seems to be the case, that's probably a
>good indication that it's a more obvious name.
For heaven's sake. If it's being "adopted" it must not be
standard. If it's being adopted primarily by non-native
speakers that's very strong evidence that it's not standard.
Assuming that your comments are supposed to have some
relevance to the things I've said that you seem to be disputing:
This is getting very silly. Some years ago we had a grad student
here who never could get straight the difference in meaning
between "an apple" and "the apple". If I were insisting that
the two do not mean the same thing in standard English you'd
take this non-native speaker's speech as evidence that in fact
there _is_ no difference?
Timothy Murphy
> The "reference" for what is and
> what is not standard mathematical English is the body
> of mathematics written in English.
That's a very silly definition of good English usage,
whether in mathematics or anywhere else.
You seem to have set yourself up as some sort of authority
on English usage.
Unfortunately your English does not strike me
as particularly elegant.
Your sentences are much too long.
You might study George Orwell's 6 Rules for Writing English
to some advantage.
Denis Feldmann
> David C. Ullrich wrote:
>
>> The "reference" for what is and
>> what is not standard mathematical English is the body
>> of mathematics written in English.
>
> That's a very silly definition of good English usage,
> whether in mathematics or anywhere else.
Of "good" usage, perhaps, but of (correct) usage, what other definiton
would you like?
>
> You seem to have set yourself up as some sort of authority
> on English usage.
As is any native speaker (see Chomsky's work, for instance)
> Unfortunately your English does not strike me
> as particularly elegant.
> Your sentences are much too long.
Really, this thread is degenrating. How low will you stoop instead of
admitting that, on this *very minor* point, you were wrong ?
> You might study George Orwell's 6 Rules for Writing English
> to some advantage.
Somehow,I feel that "Freedom is the freedom to say that two plus two is
four ; this granted, the rest follows" would be a more appropriate
Orwell's quotation for this forum (if not for this thread)
>
Timothy Murphy
>>> The "reference" for what is and
>>> what is not standard mathematical English is the body
>>> of mathematics written in English.
>>
>> That's a very silly definition of good English usage,
>> whether in mathematics or anywhere else.
>
> Of "good" usage, perhaps, but of (correct) usage, what other definiton
> would you like?
So you believe correct usage is a matter of democracy?
Do you also apply this to music?
If all the banks give out loans like water
does that make it correct banking usage?
>> You seem to have set yourself up as some sort of authority
>> on English usage.
>
> As is any native speaker (see Chomsky's work, for instance)
Sorry, are you now setting Chomsky before us
as an authority on English usage?
I find his political writing almost unreadable.
Only yesterday I waded through a double-page article
in my newspaper by the great man,
from which the only thing I was quite clear about
was that Chomsky is not very fond of Bush.
galathaea
> Denis Feldmann wrote:
> >>> The "reference" for what is and
> >>> what is not standard mathematical English is the body
> >>> of mathematics written in English.
>
> >> That's a very silly definition of good English usage,
> >> whether in mathematics or anywhere else.
>
> > Of "good" usage, perhaps, but of (correct) usage, what other definiton
> > would you like?
>
> So you believe correct usage is a matter of democracy?
> Do you also apply this to music?
> If all the banks give out loans like water
> does that make it correct banking usage?
>
> >> You seem to have set yourself up as some sort of authority
> >> on English usage.
>
> > As is any native speaker (see Chomsky's work, for instance)
>
> Sorry, are you now setting Chomsky before us
> as an authority on English usage?
> I find his political writing almost unreadable.
> Only yesterday I waded through a double-page article
> in my newspaper by the great man,
> from which the only thing I was quite clear about
> was that Chomsky is not very fond of Bush.
is this argument still going on?
i thought it was obvious what was meant by "in x algebra"
just as it is clear what one means by "in lambda calculus"
an algebra is a computational system
and there are theorems provable in that system
and...
"in boolean algebra the law of excluded middle holds"
anyway
there is no "correct" usage in any language
and chomsky never said anything that pushes enforcements
onto some authority or democracy
his "manufacturing consent"
though
is a great analysis of how a private and free media
can develop natural dynamics
that push it's agenda towards corporatism
due various forces of
advertisements and marketing as income source
laziness to pursue press releases as news
...
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
Michael Press
Timothy Murphy <t...@maths.tcd.ie> wrote:
> David C. Ullrich wrote:
>
> > The "reference" for what is and
> > what is not standard mathematical English is the body
> > of mathematics written in English.
>
> That's a very silly definition of good English usage,
> whether in mathematics or anywhere else.
>
> You seem to have set yourself up as some sort of authority
> on English usage.
> Unfortunately your English does not strike me
> as particularly elegant.
> Your sentences are much too long.
> You might study George Orwell's 6 Rules for Writing English
> to some advantage.
How do you come to hold up this here George Orwell
as an authority on writing English? I bet he's some
kind of foreigner.
--
Michael Press
Denis Feldmann
> Denis Feldmann wrote:
>
>>>> The "reference" for what is and
>>>> what is not standard mathematical English is the body
>>>> of mathematics written in English.
>>> That's a very silly definition of good English usage,
>>> whether in mathematics or anywhere else.
>> Of "good" usage, perhaps, but of (correct) usage, what other definiton
>> would you like?
>
> So you believe correct usage is a matter of democracy?
No, of usage. And you didn't answer my question (as for linguistic, see
"normative vs. descriptive").
> Do you also apply this to music?
To folk music (as opposed to savant music), yes
> If all the banks give out loans like water
> does that make it correct banking usage?
Not exactly the right word, but yes, it defines banking present usage
(and it would be good for the public to understand this as soon s
possible, instead of seeing it as an nomaly)
>
>>> You seem to have set yourself up as some sort of authority
>>> on English usage.
>> As is any native speaker (see Chomsky's work, for instance)
>
> Sorry, are you now setting Chomsky before us
> as an authority on English usage?
No, as an authority in linguistic (but Saussure would have done as well)
> I find his political writing almost unreadable.
Yes, but you have to admit that your text here is not of a very good
mathematical quality. Anyway, I will come back (in your threads) when
you have something mathmatical to tell. For linguistic (and politics,
and economy), I usually go elsewhere
David C. Ullrich
wrote:
>David C. Ullrich wrote:
>
>> The "reference" for what is and
>> what is not standard mathematical English is the body
>> of mathematics written in English.
>
>That's a very silly definition of good English usage,
>whether in mathematics or anywhere else.
I didn't say "good", I said "standard".
>You seem to have set yourself up as some sort of authority
>on English usage.
Does it seem that way to you?
You're a funny guy. You really _would_ prefer
sounding illiterate some day when you're giving
a talk or writing a paper to being corrected here.
(Cf. Phil's currently last word on the topic, to
the effect that it seems the usage in question
is being adopted by non-native speakers...
there's nothing wrong with being a non-native
speaker, but most native speakers wouldn't
want to sound like one.)
Or so it "seems". Whether I'm an authority on
usage or not, your usage of "in Banach algebra"
is simply wrong.
>Unfortunately your English does not strike me
>as particularly elegant.
>Your sentences are much too long.
Supposing that were so (curiously, no editor has
ever had any complaint about this, but never mind),
that has no bearing on the question of whether
you're using the phrase "Banach algebra" correctly.
It does have some bearing on various strawmen.
For example, if I'd claimed that my English was
in fact particularly elegant it would have some
relevance to that. If I'd claimed that the fact
that I'm an authority on English usage _showed_
that I was right about "Banach algebra" it would
have some bearing on _that_.
But I never claimed to have a proof that I'm
right here. I simply (and correctly) informed you
that the phrase is not used that way. If you didn't
want to seem ridiculous you would have ignored
the comment or thanked me for pointing it out.
>You might study George Orwell's 6 Rules for Writing English
>to some advantage.
Right. He was talking about technical writing, right?
Hmm. My favorite is number five, "Never use a foreign phrase, a
scientific word, or a jargon word if you can think of an everyday
English equivalent."
But I don't see anything on the list that would indicate
that your use of "Banach algebra" is correct.
Timothy Murphy
> And you didn't answer my question (as for linguistic, see
> "normative vs. descriptive").
As far as I can see, the only question you asked me was this:
------------------------------------
>> The "reference" for what is and
>> what is not standard mathematical English is the body
>> of mathematics written in English.
>
> That's a very silly definition of good English usage,
> whether in mathematics or anywhere else.
Of "good" usage, perhaps, but of (correct) usage, what other definiton
would you like?
------------------------------------
My answer would be: an accepted authority on mathematical usage,
such as Bourbaki.
Incidentally, why _did_ you bring Chomsky into this?
Do you think he provides a shining example of English usage?
Strange.
Phil Carmody
> wrote:
>
>>David C. Ullrich wrote:
>>
>>> The "reference" for what is and
>>> what is not standard mathematical English is the body
>>> of mathematics written in English.
>>
>>That's a very silly definition of good English usage,
>>whether in mathematics or anywhere else.
>
> I didn't say "good", I said "standard".
>
>>You seem to have set yourself up as some sort of authority
>>on English usage.
>
> Does it seem that way to you?
>
> You're a funny guy. You really _would_ prefer
> sounding illiterate some day when you're giving
> a talk or writing a paper to being corrected here.
> (Cf. Phil's currently last word on the topic,
My last words on the topic were, and still are, "what standard?"
You've failed to answer that repeatedly, by which I mean you've
failed to profide a reference to the standardisation document
for English. Failing repeatedly to answer a simple question is
a trait normally associated with the loons on this group.
Timothy Murphy
>>You seem to have set yourself up as some sort of authority
>>on English usage.
>
> Does it seem that way to you?
Yes.
If you say on some subject, "That is right; and this is wrong",
without providing any evidence to back up your opinion,
then you _are_ setting yourself up as an authority on the subject.
Incidentally, this thread started when I asked
a straightforward question about Pontryagin duality,
namely did anyone have a simple proof that every character
of the dual group G* of a locally compact abelian group
arose from element of G.
Instead of answering the question,
you wrote an offensive criticism of the language in which
the question was posed.
As it happened, I found a reasonably simple proof
in Reiter & Stegeman, "Classical Harmonic Analysis
and Locally Compact Groups".
David Bernier
> Denis Feldmann wrote:
>
>> And you didn't answer my question (as for linguistic, see
>> "normative vs. descriptive").
>
> As far as I can see, the only question you asked me was this:
> ------------------------------------
>>> The "reference" for what is and
>>> what is not standard mathematical English is the body
>>> of mathematics written in English.
>> That's a very silly definition of good English usage,
>> whether in mathematics or anywhere else.
>
> Of "good" usage, perhaps, but of (correct) usage, what other definiton
> would you like?
> ------------------------------------
>
> My answer would be: an accepted authority on mathematical usage,
> such as Bourbaki.
In French, the area "Boolean algebra" is translated as
"l'algebre booleenne" or "l'algebre de Boole".
For some object that is "a Boolean algebra", it would be:
"une algebre de Boole".
The usage "Banach algebra had its beginnings in [etc.]" would be
translated as: "L'algebre de Banach a eue ses debuts [etc.] ".
Personally, I would say or write: "une algebre de Banach
est par definition un espace de Banach reel ou complexe
muni d'une structure d'anneau tel que [etc.]",
but not: "Gelfand a fait des travaux importants
dans le domaine de l'algebre de Banach."
There is too much by Bourbaki for me to know whether
"he" writes like that or not.
David Bernier
David C. Ullrich
<thefatphi...@yahoo.co.uk> wrote:
>David C. Ullrich <dull...@sprynet.com> writes:
>> On Mon, 13 Oct 2008 13:30:05 +0100, Timothy Murphy <t...@maths.tcd.ie>
>> wrote:
>>
>>>David C. Ullrich wrote:
>>>
>>>> The "reference" for what is and
>>>> what is not standard mathematical English is the body
>>>> of mathematics written in English.
>>>
>>>That's a very silly definition of good English usage,
>>>whether in mathematics or anywhere else.
>>
>> I didn't say "good", I said "standard".
>>
>>>You seem to have set yourself up as some sort of authority
>>>on English usage.
>>
>> Does it seem that way to you?
>>
>> You're a funny guy. You really _would_ prefer
>> sounding illiterate some day when you're giving
>> a talk or writing a paper to being corrected here.
>> (Cf. Phil's currently last word on the topic,
>
>My last words on the topic were, and still are, "what standard?"
>You've failed to answer that repeatedly,
That's not true and you know it.
>by which I mean you've
>failed to profide a reference to the standardisation document
>for English. Failing repeatedly to answer a simple question is
>a trait normally associated with the loons on this group.
Yes. But failing to answer a question is not the same
thing as failing to do _what you mean by_ answering
a question.
>Phil
David C. Ullrich
wrote:
>David C. Ullrich wrote:
>
>>>You seem to have set yourself up as some sort of authority
>>>on English usage.
>>
>> Does it seem that way to you?
>
>Yes.
>If you say on some subject, "That is right; and this is wrong",
>without providing any evidence to back up your opinion,
>then you _are_ setting yourself up as an authority on the subject.
If you insist, then yes, I'm an authority on whether on not
your usage of "Banach algebra" is standard.
It's not possible to prove that a certain usage does not occur
without inspecting every document on the topic. And the
fact that a usage occurs rarely doesn't show that it's correct
or standard. Everything I've ever read on Banach algebras
is evidence that I'm right - no, I haven't provided citations
to everything I've ever read. Everything _you've_ ever
read on the topic is _also_ evidence - the fact that you
refuse to admit this doesn't mean it's not so.
On the other hand, you've failed to provide a single
instance where the usage in question is actually used.
>Incidentally, this thread started when I asked
>a straightforward question about Pontryagin duality,
>namely did anyone have a simple proof that every character
>of the dual group G* of a locally compact abelian group
>arose from element of G.
>Instead of answering the question,
>you wrote an offensive criticism of the language in which
>the question was posed.
The "instead" is not right - I also addressed the question.
I didn't give the answer you wanted.
Regarding "offensive", if you really find
'Not that it matters, but "go into Banach
algebra" is wrong, like "go into group"
would be wrong. There's no topic or technique
called "Banach algebra", rather _a_ Banach
algebra is a certain sort of object...'
offensive that's simply pathetic.
>As it happened, I found a reasonably simple proof
>in Reiter & Stegeman, "Classical Harmonic Analysis
>and Locally Compact Groups".
David C. Ullrich
Timothy Murphy
>>Incidentally, this thread started when I asked
>>a straightforward question about Pontryagin duality,
>>namely did anyone have a simple proof that every character
>>of the dual group G* of a locally compact abelian group
>>arose from element of G.
>>Instead of answering the question,
>>you wrote an offensive criticism of the language in which
>>the question was posed.
>
> The "instead" is not right - I also addressed the question.
> I didn't give the answer you wanted.
Please remind me of your answer.
I missed it.
David C. Ullrich
wrote:
>David C. Ullrich wrote:
>
>>>Incidentally, this thread started when I asked
>>>a straightforward question about Pontryagin duality,
>>>namely did anyone have a simple proof that every character
>>>of the dual group G* of a locally compact abelian group
>>>arose from element of G.
>>>Instead of answering the question,
>>>you wrote an offensive criticism of the language in which
>>>the question was posed.
>>
>> The "instead" is not right - I also addressed the question.
>> I didn't give the answer you wanted.
>
>Please remind me of your answer.
>I missed it.
That's patently offensive.