Questions on functional analysis
Ashok. R
I really really want to learn about functional analysis and I am trying
really really hard - and I have a few questions...
1. Why is closure so important and how is it different from completenss?
If a set is closed, does it necessarily imply that it is also complete?
2. Whats is the difference between a Cauchy sequence and a convergent
sequence? Can anyone give me an example of a cauchy sequence that is not
convergent?
3. Quotient space: This is the stuff that I am really struggling with. I
just cannot understand the concept of a Quotient space. Can anyone give
me an easy to follow explanation with good examples?
4. Is there an easy to follow book to learn functional analysis? I am
using Luenberger's "Optimization by vector space methods" and it is not
exactly easy to read. I need some examples, stuff that I can visualize.
5. The Null space of a matrix, N(A): Is there a good way to show that it
is complete? What would be a good cauchy sequence for N(A) ?
6. Are all complete spaces Banach spaces?
Merci Beaucoup and please forgive my ignorance if there are fundamental
errors in my questions.
Ashok.R - Starving Graduate Student (Please feed me!)
http://www.dartmouth.edu/~ashokr/ashokr
Zdislav V. Kovarik
Ashok. R <navi...@dartmouth.edu> wrote:
>Greetings.
>
>
>I really really want to learn about functional analysis and I am trying
>really really hard - and I have a few questions...
>
Looks like no takers, so I'll give it a try.
>1. Why is closure so important and how is it different from completenss?
>If a set is closed, does it necessarily imply that it is also complete?
I presume you mean closure as a property ("closedness") in contrast with
closure as operation.
Just scan theorems in your text that assume closedness, and find out which
of them would turn false if you drop that assumption. (A nice series of
comprehension exercises.) Two important cases:
In metric topology, let a set S be closed. Then every point not in S has
a positive distance from S. This is quite convenient. If S is not closed,
examples exist of x being at zero distance from S but not in S. (Take S as
the reciprocals of positive integers, and x = 0.)
If a linear subspace of a normed space is closed then the induced
seminorm on the quotient space is a norm (quite a desirable property).
And conversely: if the subspace is not closed, the induced seminorm fails
to be a norm.
And about completeness vs. closure: Being closed is much easier in a way
than being complete.
There must be a theorem in the book saying this about metric spaces:
If a space X is complete then it is closed in every bigger space Y if
subspace metric from Y is used on X.
And:
If a space X is complete then its closed subspaces are exactly its
complete subspaces.
For example, the space of polynomials from the example below is not
complete, but: It is closed in itself (of course, every space is closed in
itself!), and some of its closed subspaces are complete (the finite
dimensional ones, for example), and others are incomplete (e.g. the set of
all even poynomials).
>2. Whats is the difference between a Cauchy sequence and a convergent
>sequence? Can anyone give me an example of a cauchy sequence that is not
>convergent?
>
A one-way implication in general cases: if convergent then Cauchy.
A two-way implication in complete spaces (read the definition again).
Surroundings: Take V to be the space of polynomial functions, normed by
maximum of absolute value over [-1,1]. This is a non-closed linear
subspace of C([-1,1]).
Example: Take the n-th polynomial to be the n-th partial sum of the power
series for e^x (starting with n=0):
p_n(x) = 1 + x/1! + ... + x^n/n!
Show that {p_n} is Cauchy but not convergent (it converges to e^x, which
is not a polynomial. Why? (Observe the derivatives.)).
>3. Quotient space: This is the stuff that I am really struggling with. I
>just cannot understand the concept of a Quotient space. Can anyone give
>me an easy to follow explanation with good examples?
Example: Everyone who went through Calculus I with a good teacher
remembers that on an interval, the indefinite integral of a continuous
function is actually a class of functions, any one of them plus a constant
function.
So, for example, the indefinite integral of sin(x) is the set
{-cos(x)+C: C a constant}.
Now from the definitions, this set is a coset -cos(x)+P(0) where P(0) is
the set of all constant functions.
More generally, the result of applying the operation of indefinite
integration to the space of all continuous functions on R (for simplicity)
is the quotient space:
(continuously differentiable functions) / P(0)
(Casually speaking, you consider every constant to be just as good as
zero, because if you differentiate the integral ...)
Going higher: if you integrate indefinitely every continuous function 6
times, the result will be
(six times continuously differentiable functions) / P(5)
where P(5) is the space of all polynomials of degree 5 or less. Why? (If
you and your neighbor integrate correctly the same function six times,
each time picking your own constant, you might get different answers, but
the difference of your answers should be a polynomial from P(5), shouldn't
it?)
A more general example: Suppose L is a linear transformation of a space X
onto Y. Then we can, casually speaking, consider x just as good as y if
L(x) = L(y). (Think of a linear differential equation L(x)=f and
"particular solutions" x and y of it.)
If you denote N the nullspace of L (the set of solutions of the
homogeneous equation), then the particular solutions can be sorted out as
cosets of X/N .
>4. Is there an easy to follow book to learn functional analysis? I am
>using Luenberger's "Optimization by vector space methods" and it is not
>exactly easy to read. I need some examples, stuff that I can visualize.
Luenberger's book (I remember it quite well) was not at all intended to be
an introduction to functional analysis. Try Rudin, or Dieudonne
(Introduction to Modern Analysis), od Schechter. Others may suggest easier
ones.
>5. The Null space of a matrix, N(A): Is there a good way to show that it
>is complete? What would be a good cauchy sequence for N(A) ?
>
(Comprehension problem: You don't pick "a good Cauchy sequence" to prove
completeness: by definition, you check every, and that means every, Cauchy
sequence -- unless you have a previously proved theorem to help you avoid
that. ) You do look for a "special" Cauchy sequence if you want to prove
***incompleteness***. Got it? (I was avoiding judging Cauchy sequences :-)
Back to matrices: The matrix A has a domain R^n and target R^m (or complex
analogues). These spaces are known to be complete, the matrix
transformation is continuous, so the nullspace is closed ... justify this
and finish the proof yourself (read my previous remarks; hints are there).
>6. Are all complete spaces Banach spaces?
All complete normed linear spaces, yes. That's the definition.
Cheers, ZVK(Slavek).
David C. Ullrich
> Greetings.
>
> I really really want to learn about functional analysis and I am trying
> really really hard - and I have a few questions...
>
> 1. Why is closure so important and how is it different from completenss?
> If a set is closed, does it necessarily imply that it is also complete?
I'm surprised ZK didn't say the following explicitly - it's certainly
implicit in what he said, but it's also in the book: In fact, although
the two concepts are certainly related, they really apply to different
things altogether. A metric space can be complete or not, while it's
a _subset_ of something that's closed or not. You could say X is
complete or X is not complete, but saying "X is closed" doesn't
even make sense, strictly speaking: you say that "X is a closed
subset of Y" (and _whether_ that's true depends on both X and Y;
the very same X can be a closed subset of Y but not a closed
subset of Z).
Of course people say just "X is closed" all the time. But they
mean that "X is a closed subset of Y", where the value of Y is clear
from the context. Otoh "X is complete" is complete as it stands.
Dave L. Renfro
>I am trying really really hard - and I have a few questions...
Zdislav V. Kovarik has already addressed your
questions, so I'll just add a few remarks to
supplement his.
>1. Why is closure so important and how is it different
>from completenss? If a set is closed, does it necessarily
>imply that it is also complete?
Note that you don't have to have a limit in hand
in order to check that a sequence is a Cauchy
sequence. Thus, if you know the space is complete,
then you don't have to have a candidate for a limit
in order to check that a sequence converges or not.
>4. Is there an easy to follow book to learn functional
>analysis? I am using Luenberger's "Optimization by vector
>space methods" and it is not exactly easy to read. I need
>some examples, stuff that I can visualize.
The best book I can think of, by far, for what will
help you at this point is
Erwin Kreyszig, INTRODUCTORY FUNCTIONAL ANALYSIS
WITH APPLICATIONS, John Weley and Sons, 1978, 688 pages.
[QA 320.K74]
Another text, at a slightly higher level, but which is
filled with many neat examples and applications
[e.g. normal families (p. 35), the metric space of
non-empty compact convex subsets of R^2 under the
Hausdorff metric (p. 41), Banach limits (p. 64),
condensation of singularities (p. 79), summability
methods (p. 86), some ergodic theory (p. 148 and p. 171),
the Baire-typical function in L^1[0,1] diverges
almost everywhere (p. 155), given any modulus of
continuity \phi, then the Baire-typical continuous
function f:[0,1] ---> reals has the following property:
{x in [0,1] : f(x) = g(x)} has measure zero for every
function g(x) with modulus of continuity \phi (p. 158),
a Banach space is a Hilbert space if and only if
the parallelogram law holds (p. 168), the isoperimetric
theorem (p. 178), the Muntz theorem (p. 179), universal
series (p. 215), Banach lattices (p. 236), etc.],
is
Casper Goffman and George Pedrick, FIRST COURSE IN
FUNCTIONAL ANALYSIS, 2'nd edition, Chelsea Publishing
Company, 1983, 284 pages. [sorry ... my copy doesn't
list the library of congress call number on the title
pages]
Kreyszig is weak on quotient spaces. However, for your
purposes, this topic is essentially the same that you'll
find in linear algebra texts. Probably any author of
a functional analysis text will assume that you've seen
quotient spaces in linear algebra (I'm excluding the
Freshmean/Soph. level linear algebra courses populated
almost entirely by non-math majors; I have in mind the
course one typically takes as a Junior or Senior), and
perhaps also in an analysis course (quotients of metric
spaces: e.g. Hoffman's ANALYSIS IN EUCLIDEAN SPACES)
and/or a topology course, and so the treatment is likely
to be brief. I suggest looking over several
(advanced) undergraduate level linear algebra texts.
These are likely to be the best sources on quotient
spaces for your needs.