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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?

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It seems that the content of Riesz representation theorem is the answer to most of your questions(v1). – Qmechanic Feb 28 '12 at 1:09
To go along with the above, I think the Hilbert space metric handle's going between vectors and their linear functionals, although not much thought is given to the metric since everybody and their brother just uses the inner product. – kηives Feb 28 '12 at 3:49
up vote 2 down vote accepted

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.

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