Riesz Theorem and the true meaning of bras in Dirac notation

Prologue
It is time to start the obligatory research blog! When I put something on paper, it stays longer in my arsenal of usable concepts. And fastest way of remembering something is reading my previous notes about it. Hence, it is a good idea to write about the new concepts that I learned, and about the problems I'm working on. I hope this blog will last long and is not going to fall into oblivion, in the realm of blogs which had been started on a whim and slowly fades away in time with the increasing lack of enthusiasm and then eternally forgotten.

Ballentine's Quantum Book
Today's "look what I learned today" section is about the "bra"s in the Dirac notation. I started to read Ballentine's "Quantum Mechanics: A Modern Development". My research problem (about which I will talk later) involves joint probabilities in quantum mechanics. While searching about it on the web I came up to his book, read some pages on Google Books and liked it very much.

The book uses rigorous mathematics, and spends enough pages on fundamentals. (When I am left alone, I can't go further but I tend to go deeper, think about the details of the concepts that I already know.) Just in the first chapter, I learned where the bras come from, what the rigged Hilbert space is and some probability theory. Thanks Prof. Ballentine! As a graduate student, I do not want that anything is kept hidden from me for pedagogical reasons. I am brave enough to confront the rigged version of Hilbert spaces or to call Dirac delta a distributions, not a function. :-)

The Dual Space of Linear Functionals
I thought that I know the Dirac notation. Unfortunately, that was an illusion. Let me share my enlightenment with you.

Start with a linear vector space $V$, with an inner product $(\psi, \phi) = c$ defined on it. Elements of $V$ are symbolized with kets: $| \psi \rangle$. $(|\psi\rangle, |\phi\rangle) = c$. A ket can be a column vector, or a function etc.

Now define a dual space of linear functionals on $V$. What is a functional? Operators are mathematical objects which maps "vectors" to "vectors". (double quotes to indicate abstractness of the vectors, the kets). They take a vector and give a vector in return: $A:\psi \rightarrow \phi$, which is shown as: $A\psi=\phi$. Functionals are objects which maps vectors to numbers. They take a "vector" and give a number: $F:\psi \rightarrow c$, maybe shown as: $F \left\{ \phi \right\} = c$. For example, the norm of a vector is a functional (although not linear).

What makes an operator linear is the property that the equation $A(\alpha\psi+\beta\phi) = \alpha A \psi + \beta A \phi$ holds for every $\psi$ and $\phi$. Similarly, a functional is linear when  $F \left\{ \alpha \psi + \beta \phi \right\} = \alpha F \left\{ \psi \right\} + \beta F \left\{ \phi \right\}$ (1)

If $F_1 \left\{ \phi \right\} + F_2\left\{ \phi \right\} = \left(F_1 + F_2\right)\left\{ \phi \right\} = F_3 \left\{ \phi \right\}$, which means that addition of two functionals (two elements of the dual space) gives another element of the dual space, the dual space is closed under addition, and is itself a vector space, $V'$.

Riesz Theorem
Riesz Theorem says that there is a one to one correspondence (isomorphism) between elements of $V$ and $V'$. The operation of an arbitrary linear functional (from  $V'$) on the elements of $V$ can be imitated by an inner product of elements from $V$. $F \left\{ \phi \right\} = (f, \phi)$, where $f$ is fixed for an $F$, and $\phi$ is arbitrary.

Dirac assumed this isomorphism between $f$ and $F$. Riesz proved it, hence assumption is not necessary. Say $\left\{\phi_n\right\}$ is an orthonormal basis. Any vector in $V$ can be expanded on that basis. $\psi = \sum_n c_n \phi_n$. According to (1), when a linear functional operates on $\psi$, $F\left\{\psi\right\} = F\left\{ \sum_n c_n \phi_n \right\} = \sum_n c_n F\left\{\phi_n\right\}$. Now, we have to find a vector $f$, of which inner product with the basis vectors, has the same effect as the functional on that basis vector.

$f = \sum_m F\left\{\phi_m\right\}^* \phi_m$ does the job. $\left(f,\psi \right) = \left(  \sum_m F\left\{\phi_m\right\}^* \phi_m, \sum_n c_n \phi_n \right)$ $= \sum_m F\left\{\phi_m\right\} \sum_n c_n \left( \phi_m, \phi_n \right) = \sum_m \sum_n F\left\{\phi_m\right\} c_n \delta_{mn}$ $= \sum_n c_n F\left\{\phi_n\right\} = F\left\{\psi\right\}$. There is a unique $f$ in $V$ for each $F$ in $V'$. Hence one can use these symbols interchangeably.

In Dirac notation, the number that the functional $F$ gives, if it takes the vector $\phi$ as its argument, $F\left\{ \phi \right\}$, is shown as $\langle F | \phi \rangle$. For an $f$ defined as in the previous paragraph, $\langle F | \phi \rangle = \left(f, \phi\right)$. Thanks to the uniqueness of $f$ due to one-to-one correspondence between $f$ and $F$, we use the symbol $F$ for both the linear functional in $V'$ and the isomorphic vector in $V$. $\left(f, \phi\right) \equiv \left(F, \phi\right)$ And we get
$$\langle F | \phi \rangle = \left(F, \phi\right)$$
Hence the braket can be thought as merely a new notation for inner product. But the real motivation behind it was that the bras are linear functionals on the ket space.