Hilbert Schmidt operators as an ideal in operators. Let $H$ be a Hilbert space. For $\{e_n\}$ an orthonormal basis of $H$, we call $T\in B(H)$, a Hilbert Schmidt operator if 
$ \|T\|_2^2:=\sum_n \|Te_n\|^2 <\infty.$
I have seen somewhere before that
$
\|ST\|_2 \leq \|S\|_{B(H)} \|T\|_2
$
for all $S\in B(H)$ arbitrary. In other words, the space of Hilbert Schmidt is an ideal in $B(H)$. Where can I find this? Where can I find it for Matrices?
 A: I don't have a reference, but this is easy to show directly: we have 
$$ \| ST \|_{HS}^2 = \sum_{n = 1}^{\infty} \| STe_n \|^{2} \leq \| S\|^2 \sum_{n = 1}^{\infty} \| Te_{n} \|^{2} = \| S \|^2 \| T \|_{HS}^2,$$ where $\| \cdot \|_{HS}$ is the Hilbert-Schmidt norm. This, by itself, however is not enough to prove that the set of Hilbert-Schmidt operators form an ideal - assuming you mean two-sided ideal, you also need to show, with $S$ and $T$ as above, that $TS$ is Hilbert-Schmidt, and that the sum of two Hilbert-Schmidt operators is again Hilbert-Schmidt (along with the perfectly obvious statement that the zero-operator is Hilbert-Schmidt). The latter is easy, 
$$ \| S + S' \|_{HS}^2 = \sum_{n = 1}^{\infty} \| (S + S')e_n \|^2 \leq 
\sum_{n = 1}^{\infty} (\| Se_n \|^2 + \| S'e_n \|^2) = \|S\|_{HS}^2 + \| S'\|_{HS}^2 < \infty $$
if $S$ and $S'$ are Hilbert-Schmidt, but I expect the former to be more difficult.  
Edit: The result follows from the fact that the set of Hilbert-Schmidt operators is $^*$-closed, i.e. if $T$ is H-S, then so is $T^*$. The proof is easy: if $S \in B(H)$ and $T$ is H-S, then $(TS)^* = S^* T^*$ is H-S by the above, and hence $TS$ is H-S also. 
Another edit: I'll add that the fact that the class of H-S operators is $^*$-closed follows from the fact that 
$$ \| S \|_{HS} = \| S^* \|_{HS}. $$
This is a (small) part of exercise 17c) of section XVIII.9 in Real and Functional Analysis by Serge Lang, so it shouldn't be too difficult to show (however, at the moment, I have no idea how to do it; maybe a sign that I need to revisit some of this stuff).  
A: $$
\sum_n\|S\,T\,e_n\|^2\le\sum_n\|S\|_{B(H)}^2\,\|T\,e_n\|^2=\|S\|_{B(H)}^2\,\|T\|_2^2
$$
A: On the sum of Hilbert-Schmidt operators, "$\|(S+S')e_n\|_H^2 \leq \|Se_n\|_H^2 + \|S'e_n\|_H^2$" is a little questionable! I propose the following approach instead:
\begin{align*}
\|S+S'\|_{HS} &= \bigg[ \sum_{n=0}^{\infty} \|(S+S')e_n \|_H^2 \bigg]^{\frac{1}{2}} 
\leq \bigg[ \sum_{n=0}^{\infty} \bigg( \|Se_n\|_H + \| S'e_n \|_H \bigg)^2 \bigg]^{\frac{1}{2}} &\\ 
&\leq \bigg[ \sum_{n=0}^{\infty} \|Se_n\|_H^2 \bigg]^{\frac{1}{2}} + 
\bigg[ \sum_{n=0}^{\infty} \|S'e_n\|_H^2 \bigg]^{\frac{1}{2}}
= \|S\|_{HS} + \|S'\|_{HS} < \infty
\end{align*}
The first inequality is the triangle inequality. The second inequality is Minkowski's inequality for infinite sums, where we view $(\|Se_n\|_H)_n$ and $(\|S'e_n\|_H)_n$ as sequences of real numbers. (working in a real Hilbert space)
