= Hilbert-Schmidt operators form a Hilbert space
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{title2=$\mathcal S_2(X)$}
= Completeness of the Hilbert-Schmidt norm
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{synonym}
On a <Hilbert space> $X$ with a countable <Hilbertian basis> $(e_k)$, the <Hilbert-Schmidt operators> form a <Hilbert space> under $\langle S,T\rangle=\sum_k\langle Se_k,Te_k\rangle$, using the same linearity convention as the <inner product> on $X$. The <Cauchy-Schwarz inequality> proves convergence of this sum. The finite-sum <triangle inequality> and <Cauchy-Schwarz inequality> give $\|T\|\leq\|T\|_2$. Thus a <Cauchy sequence> $(T_n)$ in <Hilbert-Schmidt norm> converges in <operator norm> to a <bounded linear operator> $T$. For $n,m\geq N$ with $\|T_n-T_m\|_2\leq\varepsilon$, taking $m\to\infty$ in each finite column sum gives $\sum_{k=1}^K\|(T_n-T)e_k\|^2\leq\varepsilon^2$ for every $K$. Taking the supremum over $K$ proves $T_n-T\in\mathcal S_2(X)$ and $\|T_n-T\|_2\leq\varepsilon$, so $T\in\mathcal S_2(X)$ and completeness follows.
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