# Bra space and adjoint vectors

If I'm not wrong, a bra, $\langle \phi_n |$, can be thought as a linear functional that when applied to a ket vector, $| \phi_m \rangle$, returns a complex number; that is, the inner product it's a linear mapping that $\xi \rightarrow \mathbb{C}$; yet, exist a bra to each ket, and in discrete basis, the reverse it's valid too.

Well, thinking in the scope of discrete basis, my question, then, is: when we take a adjoint of a vector, we go from a vector space to a "linear functional" space? That is, when we want to calculate the inner product of $|\phi_n \rangle$ with itself, we are, as a matter of fact, applying the bra associated with the vector in the same vector?

1) Whenever one has a topological vector space (TVS) $V$ over some field $\mathbb{F}$, one can construct a dual vector space $V^*$ consisting of continuous linear functionals $f:V\to\mathbb{F}$.

2) Under relative mild conditions on the topology of $V$, it is possible to turn the dual vector space $V^*$ into a TVS. One may iterate the construct of dual vector spaces, so that more generally, one may consider the double-dual vector space $V^{**}$, the triple-dual vector space $V^{***}$, etc.

3) There is a natural/canonical injective linear map $i :V\to V^{**}$. It is defined as

$$i(v)(f):=f(v),\qquad\qquad v\in V, \qquad\qquad f\in V^*.$$

4) If the map $i$ is bijective $V\cong V^{**}$, one says that $V$ is a reflexive TVS.

5) If $V$ is an inner product space (which is a particularly nice example of a TVS), then there is a natural/canonical injective conjugated linear map $j :V\to V^*$. It is defined as

$$j(v)(w):=\langle v, w \rangle ,\qquad\qquad v,w\in V.$$

Here we follow the Dirac convention that the "bracket" $\langle\cdot, \cdot \rangle$ is conjugated linear in the first entry (as opposed to a lot of the math literature).

6) Riesz representation theorem (RRT) shows that $j$ is a bijection if $V$ is a Hilbert space. In other words, a Hilbert space is selfdual $V\cong V^*$. If one identifies $V$ with the set of kets, and $V^*$ with the set of bras, one may interpret RRT as saying that there is a natural/canonical one-to-one correspondence between bras and kets.