The completeness relation from QM in terms of inner products I remember from QM that the completeness relation says
$$ \sum_{n=1}^\infty |e_n\rangle \langle e_n | = I$$
so that $\langle x\mid y\rangle =\sum_{n=1}^\infty \langle x\mid e_n\rangle \langle e_n \mid y\rangle$.
I was recently trying to prove a result on Trace operators and one calculation was 
$$\sum_{k=1}^n \langle A g_k ,  h_k \rangle = \sum_{k=1}^n \operatorname{tr}(A(g_k\otimes h_k))$$
Where apparently if $f\in X^*$ and $y\in Y$ we define $y\otimes f:X\to Y$ by $(y\otimes f)(x) = f(x)y$;
my attempt at this:
$$\sum_{k=1}^n \operatorname{tr}(A(g_k\otimes h_k)) = \sum_{k=1}^n\sum_{i=1}^\infty \langle A(g_k \otimes h_k) e_i, e_i\rangle$$
$$= \sum_{k=1}^n\sum_{i=1}^\infty \langle A(\langle e_i,h_k\rangle g_k), e_i\rangle = \sum_{k=1}^n\sum_{i=1}^\infty \langle Ag_k,e_i\rangle\langle e_i,h_k\rangle$$
So using my naive approach from quantum mechanics I just conclude that the last term is $\sum_{k=1}^n \langle A g_k ,  h_k \rangle $.
However, I feel uneasy about this because $\langle \cdot , \cdot \rangle$ is an inner product, while $\langle \cdot \mid \cdot \rangle$ is the bra-ket notation... whatever that means.
1) Can I apply the completeness relation to make my conclusion?
2) Is there a canonical relation between $\langle \cdot, \cdot \rangle$ and $\langle \cdot \mid \cdot \rangle$?
3) How can I prove the completeness relation (I believe it's an axiom in QM, but I reckon its equivalence (in functional analysis (if it exists)) is a theorem).
 A: 
Theorem: Let $H$ be a real or complex Hilbert space, and let $\{e_{\alpha}\}_{\alpha\in\Lambda}$ be an orthonormal subset of $H$. The following are equivalent:
  
  1. $\{ e_{\alpha} \}_{\alpha\in\Lambda}$ is a complete orthonormal set, meaning that the only $x\in H$ that is orthogonal to every $e_{\alpha}$ is $x=0$.
  
  2. Parseval's identity $\|x\|^2=\sum_{\alpha\in\Lambda}|\langle x,e_{\alpha}\rangle|^2$ holds for all $x \in H$.
  
  3. The identity $\langle x,y\rangle = \sum_{\alpha\in\Lambda}\langle x,e_{\alpha}\rangle\langle e_{\alpha},y\rangle$ holds for all $x,y\in H$.
  
  4. The subspace consisting of all finite linear combinations of the $e_{\alpha}$ is dense in $H$.

The conventions in Quantum provided for ket vectors $|x\rangle$. The bras are linear functionals. On a Hilbert space there is a canonical map $x \mapsto x^*$ from vectors to linear functionals given by $x^*(y)=\langle y,x\rangle$. In this way, you may treat $\langle y | x\rangle$ in the same ways as $\langle x,y\rangle$ by use of the canonical map.
