Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them

In the setting of Hilbert Triples: Standing Notation and Background, let $\mathrm{Sym}(V)$ with its norm $\lVert\cdot\rVert_{\mathrm{Sym}(V)}$ be as fixed in Hilbert Triples: Standing Notation and Background (vi), let $\mathbb{N}$ be the set of natural numbers, and let convergence of a series of real numbers and its sum be as fixed in Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus (i). For $x\in V$ we write $|x|_{V}^{2}=|x|_{V}|x|_{V}$, and $2=1+1$.

1. (Square-summable sequences in $V$)(i) A sequence $f=(f_{k})_{k\in\mathbb{N}}$ in $V$ is square-summable in $V$ if the series $\sum_{k=1}^{\infty}|f_{k}|_{V}^{2}$ converges. Its sum is then written

$$\sigma(f)=\sum_{k=1}^{\infty}|f_{k}|_{V}^{2}.$$

Each term $|f_{k}|_{V}^{2}$ is nonnegative, by claim 5 of Elementary Arithmetic in an Ordered Field applied to $0\le|f_{k}|_{V}$ with the nonnegative multiplier $|f_{k}|_{V}$; consequently $0\le\sigma(f)$ by claim 2 of Series of Nonnegative Real Numbers, Comparison, and the Geometric Series.

2. (The trace of a form along a square-summable sequence)(ii) Let $f=(f_{k})_{k\in\mathbb{N}}$ be square-summable in $V$ and let $X\in\mathrm{Sym}(V)$. Put $a_{k}=X(f_{k},f_{k})$ and $\mu_{k}=\lVert X\rVert_{\mathrm{Sym}(V)}\,|f_{k}|_{V}^{2}$ for $k\in\mathbb{N}$. By claim 2 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity one has $|a_{k}|\le\mu_{k}$, hence $-\mu_{k}\le a_{k}\le\mu_{k}$ by claim 6 of Properties of the Absolute Value in an Ordered Field; adding $\mu_{k}$ to each of these two inequalities, by the compatibility of the order with addition (an axiom of Ordered Field), gives

$$0\le a_{k}+\mu_{k}\le 2\mu_{k}.$$

The series $\sum_{k=1}^{\infty}\mu_{k}$ and $\sum_{k=1}^{\infty}2\mu_{k}$ converge by claim 1 of Elementary Properties of Series of Real Numbers, the first with sum $\lVert X\rVert_{\mathrm{Sym}(V)}\,\sigma(f)$; so $\sum_{k=1}^{\infty}(a_{k}+\mu_{k})$ converges by claim 3 of Series of Nonnegative Real Numbers, Comparison, and the Geometric Series, and hence so does $\sum_{k=1}^{\infty}a_{k}$, again by claim 1 of Elementary Properties of Series of Real Numbers, since $a_{k}=(a_{k}+\mu_{k})+(-1)\mu_{k}$. The trace of $X$ along $f$ is the real number

$$\mathrm{Tr}_{f}X=\sum_{k=1}^{\infty}X(f_{k},f_{k}).$$