Bra vectors in Ballentine

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Stefan Ram

Jan 30, 2023, 3:46:46 AMJan 30
In the following, I will briefly describe the Fréchet-Riesz
theorem, then explain what I think bra vectors are, and finally
say why I find an explanation of bra vectors in a Wikipedia
article and in a book confusing.

According to the Fréchet-Riesz theorem, for every vector v of
a Hilbert space H with the scalar product (., .) there exists
a one-to-one continuous and linear functional (v, .).

As far as I know, this functional (v, .) is called a bra vector
in physics and is written "<v|".

From the Wikipedia page "Bra-ket notation", September 24, 2022:

|A bra is of the form <f|. Mathematically it denotes a linear
|form f: V --> C,

Perhaps the reader already sees what I mean?

As the notation is used and as it is explained in good sources, the
linear form is <f| and not f. f is a vector from the Hilbert space
(and not the dual space)! But the article says "a linear form f:
V --> C", where it should be "a linear form <f|: V --> C" or just
"a linear form V --> C". The "f" is just wrong at this point.

Now, here in Usenet, Ballentine ("Quantum Mechanics") is always
presented as a particularly recommendable book. But even there
I see a similar problem:

|The linear functionals in the dual space V' are called
|bra vectors, and are denoted as <F|. The numerical value
|of the functional is denoted as
| F(phi)=<F|phi>.

. In "F(phi)", "F" is used as the functional. But the functional
is "<F|"; "F" is a vector from the vector space!

It is true that sometimes one can identify vectors with functionals
and use the same symbol for both. But Dirac notation was created
precisely to distinguish "<v|" from "|v>", and when a new notation
is being introduced, one should be as clear as possible.

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