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.