Proof of Elementary Properties of the Trace of a Form along a Square-Summable Sequence

Each result cited is universally quantified over the data in its own statement. Throughout, $k$ and $m$ range over $\mathbb{N}$, and for $X\in\mathrm{Sym}(V)$ we write $a_{k}(X)=X(f_{k},f_{k})$, so that $\mathrm{Tr}_{f}X=\sum_{k=1}^{\infty}a_{k}(X)$, a convergent series by Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them (ii). We use repeatedly that the norm of a member $b$ of $\mathrm{Sym}(E)$, for a real inner product space $E$, is nonnegative: by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (ii) it is the greatest lower bound of a set of nonnegative reals, of which $0$ is a lower bound, so $0\le\lVert b\rVert$ by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below.

Claim 1. By Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv), $a_{k}(X+X')=a_{k}(X)+a_{k}(X')$ and $a_{k}(\lambda X)=\lambda\,a_{k}(X)$ for every $k$. The three series $\sum_{k=1}^{\infty}a_{k}(X)$, $\sum_{k=1}^{\infty}a_{k}(X')$ and $\sum_{k=1}^{\infty}a_{k}(X+X')$ converge, so claim 1 of Elementary Properties of Series of Real Numbers gives the first two identities. For the third, $0_{\mathrm{Sym}}\in\mathrm{Sym}(V)$ by claim 1 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity, and $0\,0_{\mathrm{Sym}}=0_{\mathrm{Sym}}$, both forms taking the value $0$ at every pair; so the homogeneity just proved, applied with $\lambda=0$, gives $\mathrm{Tr}_{f}0_{\mathrm{Sym}}=0\cdot\mathrm{Tr}_{f}0_{\mathrm{Sym}}=0$.

Claim 2. Suppose $X\preceq X'$. Since $f_{k}\in V$, Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iii) gives $a_{k}(X)\le a_{k}(X')$ for every $k$. Both series converge, so claim 4 of Elementary Properties of Series of Real Numbers gives $\mathrm{Tr}_{f}X\le\mathrm{Tr}_{f}X'$.

Claim 3. Put $\mu_{k}=\lVert X\rVert_{\mathrm{Sym}(V)}\,|f_{k}|_{V}^{2}$. As recorded in Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them (ii), $-\mu_{k}\le a_{k}(X)\le\mu_{k}$ for every $k$, and $\sum_{k=1}^{\infty}\mu_{k}$ converges with sum $\lVert X\rVert_{\mathrm{Sym}(V)}\,\sigma(f)$; by claim 1 of Elementary Properties of Series of Real Numbers the series $\sum_{k=1}^{\infty}(-\mu_{k})$ converges with sum $-\lVert X\rVert_{\mathrm{Sym}(V)}\,\sigma(f)$. Claim 4 of that lemma, applied twice, gives

$$-\lVert X\rVert_{\mathrm{Sym}(V)}\,\sigma(f)\ \le\ \mathrm{Tr}_{f}X\ \le\ \lVert X\rVert_{\mathrm{Sym}(V)}\,\sigma(f),$$

and claim 6 of Properties of the Absolute Value in an Ordered Field turns this into the stated bound.

Claim 4. By Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv), $I_{V}(f_{k},f_{k})=\langle f_{k},f_{k}\rangle_{V}$, which equals $|f_{k}|_{V}^{2}$ by Real Inner Product Spaces (ii). The series defining $\mathrm{Tr}_{f}I_{V}$ and $\sigma(f)$ therefore have the same terms, and hence the same sum.

Claim 5. Let $k\in\mathbb{N}$. Since $f_{k}\in V$, Hilbert Triples: Standing Notation and Background (i) gives $|f_{k}|_{H}\le|f_{k}|_{V}$, and both are nonnegative; so $|f_{k}|_{H}^{2}\le|f_{k}|_{V}^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and $0\le|f_{k}|_{H}^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field. By claim 3 of Series of Nonnegative Real Numbers, Comparison, and the Geometric Series the series $\sum_{k=1}^{\infty}|f_{k}|_{H}^{2}$ converges and its sum $\sigma_{H}(f)$ satisfies $\sigma_{H}(f)\le\sigma(f)$; and $0\le\sigma_{H}(f)$ by claim 2 of that lemma.

Let $Y\in\mathrm{Sym}(H)$. Then $Y|_{V}\in\mathrm{Sym}(V)$ by Hilbert Triples: Standing Notation and Background (vi), so $\mathrm{Tr}_{f}(Y|_{V})$ is defined, and $Y|_{V}(f_{k},f_{k})=Y(f_{k},f_{k})$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (v). By claim 2 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity, $|Y(f_{k},f_{k})|\le\lVert Y\rVert\,|f_{k}|_{H}^{2}$; the series $\sum_{k=1}^{\infty}\lVert Y\rVert\,|f_{k}|_{H}^{2}$ converges with sum $\lVert Y\rVert\,\sigma_{H}(f)$ by claim 1 of Elementary Properties of Series of Real Numbers, so the argument of claim 3, with $\lVert Y\rVert\,|f_{k}|_{H}^{2}$ in place of $\mu_{k}$, gives $|\mathrm{Tr}_{f}(Y|_{V})|\le\lVert Y\rVert\,\sigma_{H}(f)$. Finally $\lVert Y\rVert\,\sigma_{H}(f)\le\lVert Y\rVert\,\sigma(f)$ by claim 5 of Elementary Arithmetic in an Ordered Field, the multiplier $\lVert Y\rVert$ being nonnegative.

Claim 6, the tail forms. Let $m\in\mathbb{N}$. By The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple (i), $N_{m}$ is the tail form, in the sense of Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions (iii), of the orthonormal $m$-tuple with components $e_{1},\dots,e_{m}$. Write $P_{m}z=\sum_{i=1}^{m}\langle z,e_{i}\rangle_{H}e_{i}$ for the map of Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions formed with that tuple. That clause gives

$$N_{m}(z,z)=|z-P_{m}z|_{H}^{2}\quad(z\in H), \qquad 0_{\mathrm{Sym}}\preceq N_{m}\preceq I_{H}.$$

Put $\beta_{k}(m)=N_{m}|_{V}(f_{k},f_{k})$, which equals $N_{m}(f_{k},f_{k})=|f_{k}-P_{m}f_{k}|_{H}^{2}$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (v). Applying Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iii) to $0_{\mathrm{Sym}}\preceq N_{m}\preceq I_{H}$ at the vector $f_{k}$, and then claim 5,

$$0\le\beta_{k}(m)\le|f_{k}|_{H}^{2}\le|f_{k}|_{V}^{2}\qquad\text{for all }k,m\in\mathbb{N}.$$

In particular $\mathrm{Tr}_{f}(N_{m}|_{V})=\sum_{k=1}^{\infty}\beta_{k}(m)$ is nonnegative, by claim 2 of Series of Nonnegative Real Numbers, Comparison, and the Geometric Series.

Claim 6, the limit. Let $\varepsilon\in\mathbb{R}$ be positive; the order of the choices below is: first the cut-off $n$, then the index $m_{0}$.

The cut-off. The partial sums of $\sum_{k=1}^{\infty}|f_{k}|_{V}^{2}$ converge to $\sigma(f)$ by Series of Real Numbers (ii), so there is $n\in\mathbb{N}$ with

$$\sigma(f)-\sum_{k=1}^{n}|f_{k}|_{V}^{2}<\tfrac{\varepsilon}{2}.$$

The leading terms. Fix $k\in\mathbb{N}$. By Orthonormal Expansions in a Real Hilbert Space (iii) the series $\sum_{i=1}^{\infty}\langle f_{k},e_{i}\rangle_{H}e_{i}$ converges in $H$ with sum $f_{k}$; its $m$-th partial sum is $P_{m}f_{k}$, so the sequence $(P_{m}f_{k})_{m\in\mathbb{N}}$ converges to $f_{k}$ in $(H,d_{H})$, that is, $(|f_{k}-P_{m}f_{k}|_{H})_{m\in\mathbb{N}}$ converges to $0$, the distance being the norm of the difference by Real Inner Product Spaces (iii). By claim 2 of Arithmetic of Limits of Real Sequences the sequence $(\beta_{k}(m))_{m\in\mathbb{N}}$, whose terms are products of that sequence with itself, converges to $0$. Hence, by Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits (iii) applied with $N=n$ to these $n$ sequences, the sequence whose $m$-th term is $\sum_{k=1}^{n}\beta_{k}(m)$ converges to $\sum_{k=1}^{n}0$, which is $0$ by claim 3 of Properties of Finite Sums applied with $\lambda=0$. So there is $m_{0}\in\mathbb{N}$ such that

$$\sum_{k=1}^{n}\beta_{k}(m)<\tfrac{\varepsilon}{2}\qquad\text{for every }m\in\mathbb{N}\text{ with }m_{0}\le m,$$

the sum being nonnegative by claim 5 of Properties of Finite Sums.

The remaining terms. Let $m\in\mathbb{N}$, put $\mu_{k}=|f_{k}|_{V}^{2}$, and let $w_{k}$ be $\beta_{k}(m)\tfrac{1}{\mu_{k}}$ if $\mu_{k}\ne0$ and $0$ if $\mu_{k}=0$. If $\mu_{k}=0$ then $\beta_{k}(m)=0$, since $0\le\beta_{k}(m)\le\mu_{k}$; so $\mu_{k}w_{k}=\beta_{k}(m)$ in either case. Moreover $0\le w_{k}\le1$: this is clear when $\mu_{k}=0$, and otherwise $\mu_{k}$ is positive, so $\tfrac{1}{\mu_{k}}$ is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field, and multiplying $0\le\beta_{k}(m)\le\mu_{k}$ by it, by claim 5 of Elementary Arithmetic in an Ordered Field, gives $0\le w_{k}\le1$. Claim 5 of Series of Nonnegative Real Numbers, Comparison, and the Geometric Series, applied with these $\mu_{k}$ and $w_{k}$ and with $M=1$, now gives

$$0\le\sum_{k=1}^{\infty}\beta_{k}(m)-\sum_{k=1}^{n}\beta_{k}(m)\le\sigma(f)-\sum_{k=1}^{n}\mu_{k}<\tfrac{\varepsilon}{2}.$$

Conclusion. Let $m\in\mathbb{N}$ with $m_{0}\le m$. Adding the two displayed estimates gives $0\le\mathrm{Tr}_{f}(N_{m}|_{V})<\varepsilon$, whence $|\mathrm{Tr}_{f}(N_{m}|_{V})-0|<\varepsilon$ by claim 1 of Properties of the Absolute Value in an Ordered Field. As $\varepsilon$ was an arbitrary positive real, the sequence $(\mathrm{Tr}_{f}(N_{m}|_{V}))_{m\in\mathbb{N}}$ converges to $0$.