Proof of In a Diagonal Hilbert Triple whose Weights Tend to Infinity, Bounded Sequences of the Form Space Have Subsequences Converging in the Ambient Space

Each result cited below is universally quantified over the data in its own statement. We work in the settings of Real Hilbert Spaces: Standing Notation and Background and Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus, and we adopt The Real Numbers: Standing Notation and Background, which carries the elementary arithmetic, order and absolute-value facts of $\mathbb{R}$ and the manipulations of finite sums (termwise comparison, splitting the index set into disjoint parts, factoring out constants) by the finite-sum results in force by The Real Numbers: Standing Notation and Background §background.

Step 0 (Notation and the norm of $V$). For $y\in H$ and $k\in\mathbb{N}$ write $y_{k}=\langle y,e_{k}\rangle_{H}$; for the given sequence write $x_{j,k}=\langle x_{j},e_{k}\rangle_{H}$. By the standing hypotheses of The Diagonal Hilbert Triple Determined by an Orthonormal Basis and a Sequence of Weights, under which the triple of The Diagonal Hilbert Triple Determined by an Orthonormal Basis and a Sequence of Weights §triple is formed, $1\le\lambda_{k}$ for every $k$, and $V$ with $\langle\cdot,\cdot\rangle_{V}$ is the weighted coefficient subspace of The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights: $V$ is the set of those $y\in H$ for which the series $\sum_{k=1}^{\infty}\lambda_{k}y_{k}^{2}$ converges, and $\langle y,z\rangle_{V}=\sum_{k=1}^{\infty}\lambda_{k}y_{k}z_{k}$ for $y,z\in V$. By Real Inner Product Space §norm, every $y\in V$ satisfies $0\le|y|_{V}$ and

$$|y|_{V}^{2}=\langle y,y\rangle_{V}=\sum_{k=1}^{\infty}\lambda_{k}y_{k}^{2}.$$

The terms $\lambda_{k}y_{k}^{2}$ are nonnegative, so Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates gives

$$\sum_{k=1}^{n}\lambda_{k}y_{k}^{2}\le|y|_{V}^{2}\qquad\text{for every }y\in V\text{ and }n\in\mathbb{N}. \tag{0.1}$$

If moreover $|y|_{V}\le R$, then $|y|_{V}^{2}\le R^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both numbers being nonnegative. Finally, for $y,z\in H$ we have $(y-z)_{k}=y_{k}-z_{k}$ by Elementary Identities in a Real Inner Product Space §bilinear, and $|e_{k}|_{H}=1$ for every $k$ by Orthogonality, Orthogonal Complement and Orthonormal Families in a Real Inner Product Space §orthonormal, since an orthonormal basis is in particular an orthonormal sequence by Orthonormal Basis of a Real Hilbert Space §basis.

Step 1 (Bounded coordinates). Let $j,k\in\mathbb{N}$. By The Cauchy-Schwarz Inequality in a Real Inner Product Space, $|x_{j,k}|\le|x_{j}|_{H}\,|e_{k}|_{H}=|x_{j}|_{H}$, and by The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights §subspace, $|x_{j}|_{H}\le|x_{j}|_{V}\le R$. Hence $|x_{j,k}|\le R$ for all $j,k\in\mathbb{N}$.

Step 2 (A coordinatewise convergent subsequence). Apply The Diagonal Subsequence Lemma for Bounded Real Arrays to the map $a:\mathbb{N}\times\mathbb{N}\to\mathbb{R}$, $a_{m,k}=x_{m,k}$, with $R_{k}=R$ for every $k$, which is admissible by Step 1. It gives a strictly increasing sequence $(j_{i})_{i\in\mathbb{N}}$ in $\mathbb{N}$, so that $j_{1}<j_{2}<j_{3}<\cdots$, such that for every $k\in\mathbb{N}$ the sequence $(x_{j_{i},k})_{i\in\mathbb{N}}$ converges; call its limit $c_{k}$. Write $y^{(i)}=x_{j_{i}}$, so that $y^{(i)}\in V$, $|y^{(i)}|_{V}\le R$, and $(y^{(i)}_{k})_{i\in\mathbb{N}}$ converges to $c_{k}$ for every $k$.

Step 3 (Tail estimate). Let $M$ be a positive real number and $k_{0}\in\mathbb{N}$ with $M<\lambda_{k}$ for every $k\ge k_{0}$. For $n\in\mathbb{N}$ with $n\ge k_{0}$ put $F_{n}=\{k\in[n]:k_{0}\le k\}$, which is nonempty since $n\in F_{n}$. We show that for all $y,z\in V$ with $|y|_{V}\le R$ and $|z|_{V}\le R$,

$$\sum_{k\in F_{n}}(y_{k}-z_{k})^{2}\le\frac{4R^{2}}{M}. \tag{3.1}$$

For $k\in F_{n}$, Products and Sums of Weighted Square-Summable Sequences of Real Numbers §pointwise, applied with $s=y_{k}$ and $t=-z_{k}$, gives $(y_{k}-z_{k})^{2}\le2y_{k}^{2}+2z_{k}^{2}$; and since $0<M<\lambda_{k}$ and $0\le y_{k}^{2}$, we have $y_{k}^{2}\le\tfrac{1}{M}\lambda_{k}y_{k}^{2}$, and likewise for $z$. Hence $(y_{k}-z_{k})^{2}\le\tfrac{2}{M}\bigl(\lambda_{k}y_{k}^{2}+\lambda_{k}z_{k}^{2}\bigr)$ for $k\in F_{n}$. Summing over $F_{n}$, and using that the omitted terms $\lambda_{k}y_{k}^{2}$ with $k\in[n]$, $k<k_{0}$, are nonnegative, we get $\sum_{k\in F_{n}}\lambda_{k}y_{k}^{2}\le\sum_{k=1}^{n}\lambda_{k}y_{k}^{2}\le R^{2}$ by (0.1) and Step 0, and likewise for $z$. Therefore the left side of (3.1) is at most $\tfrac{2}{M}(R^{2}+R^{2})=\tfrac{4R^{2}}{M}$.

Step 4 (The subsequence is Cauchy in $H$). Fix a real $\varepsilon>0$. The choices below are made in this order: $M$, then $k_{0}$, then $\tau$, then $I$.

First put $M=\tfrac{16R^{2}}{\varepsilon^{2}}+1$, which is positive and satisfies $16R^{2}\le\varepsilon^{2}M$, hence $\tfrac{4R^{2}}{M}\le\tfrac{\varepsilon^{2}}{4}$. Second, the hypothesis on the weights gives $k'\in\mathbb{N}$ with $M<\lambda_{k}$ for every $k\ge k'$; put $k_{0}=k'+1$, so that $k_{0}\ge2$ and $M<\lambda_{k}$ for every $k\ge k_{0}$, and let $E=[k_{0}-1]$, the nonempty set of $k\in\mathbb{N}$ with $k<k_{0}$. Third, put $\theta=\tfrac{\varepsilon^{2}}{4k_{0}}>0$ and $\tau=\min\{1,\theta\}$, so that $0<\tau\le1$ and hence $\tau^{2}\le\tau\le\theta$. Fourth, for each $k\in E$ the sequence $(y^{(i)}_{k})_{i\in\mathbb{N}}$ converges to $c_{k}$, so by Limit of a Sequence of Real Numbers there is $I_{k}\in\mathbb{N}$ with $|y^{(i)}_{k}-c_{k}|<\tfrac{\tau}{2}$ for every $i\ge I_{k}$; put $I=\sum_{k=1}^{k_{0}-1}I_{k}$, a natural number with $I\ge I_{k}$ for every $k\in E$.

Now let $i,l\in\mathbb{N}$ with $i\ge I$ and $l\ge I$, put $w=y^{(i)}-y^{(l)}\in H$, so that $w_{k}=y^{(i)}_{k}-y^{(l)}_{k}$ by Step 0, and let $s_{n}=\sum_{k=1}^{n}w_{k}^{2}$. For $k\in E$ we have $|w_{k}|\le|y^{(i)}_{k}-c_{k}|+|c_{k}-y^{(l)}_{k}|<\tau$, so $w_{k}^{2}=|w_{k}|^{2}<\tau^{2}\le\theta$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. If $n<k_{0}$, then $[n]\subseteq E$ and $s_{n}\le n\theta\le k_{0}\theta=\tfrac{\varepsilon^{2}}{4}$. If $n\ge k_{0}$, then $[n]$ is the disjoint union of $E$ and $F_{n}$, and by (3.1), applied to $y=y^{(i)}$ and $z=y^{(l)}$,

$$s_{n}=\sum_{k\in E}w_{k}^{2}+\sum_{k\in F_{n}}w_{k}^{2}\le(k_{0}-1)\theta+\frac{4R^{2}}{M}\le\frac{\varepsilon^{2}}{4}+\frac{\varepsilon^{2}}{4}=\frac{\varepsilon^{2}}{2}.$$

Thus $s_{n}\le\tfrac{\varepsilon^{2}}{2}$ for every $n\in\mathbb{N}$. By Orthonormal Expansions in a Real Hilbert Space §parseval, applied to $w$, the series $\sum_{k=1}^{\infty}w_{k}^{2}$ converges with sum $|w|_{H}^{2}$, and by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion this sum is $\sup\{s_{n}:n\in\mathbb{N}\}$, which is at most $\tfrac{\varepsilon^{2}}{2}<\varepsilon^{2}$. Since $0\le|w|_{H}$ and $0<\varepsilon$, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $|w|_{H}<\varepsilon$, and by Real Inner Product Space §distance, $d_{H}(y^{(i)},y^{(l)})=|w|_{H}<\varepsilon$. As $\varepsilon$ was arbitrary, $(y^{(i)})_{i\in\mathbb{N}}$ is a Cauchy sequence in $(H,d_{H})$.

Step 5 (The limit in $H$). Since $H$ is a real Hilbert space, the metric space $(H,d_{H})$ is complete by Real Hilbert Space §hilbert. Hence there is $x\in H$ such that $(y^{(i)})_{i\in\mathbb{N}}=(x_{j_{i}})_{i\in\mathbb{N}}$ converges to $x$ in $(H,d_{H})$.

Step 6 ($x\in V$ and $|x|_{V}\le R$). For $k\in\mathbb{N}$ write $\xi_{k}=\langle x,e_{k}\rangle_{H}$, the coordinate of the limit $x$ in the notation of Step 0. Fix $k\in\mathbb{N}$. By Step 0, The Cauchy-Schwarz Inequality in a Real Inner Product Space and Real Inner Product Space §distance,

$$|y^{(i)}_{k}-\xi_{k}|=|\langle y^{(i)}-x,e_{k}\rangle_{H}|\le|y^{(i)}-x|_{H}\,|e_{k}|_{H}=d_{H}(y^{(i)},x).$$

Given $\eta>0$, Convergent Sequence in a Metric Space gives $I'\in\mathbb{N}$ with $d_{H}(y^{(i)},x)<\eta$ for every $i\ge I'$, so by Limit of a Sequence of Real Numbers the sequence $(y^{(i)}_{k})_{i\in\mathbb{N}}$ converges to $\xi_{k}$. By Arithmetic of Limits of Real Sequences §products and Arithmetic of Limits of Real Sequences §scalar, $(\lambda_{k}(y^{(i)}_{k})^{2})_{i\in\mathbb{N}}$ converges to $\lambda_{k}\xi_{k}^{2}$ for every $k$.

Now fix $n\in\mathbb{N}$. By Arithmetic of Limits of Real Sequences §sums and induction on $n$, the sequence $\bigl(\sum_{k=1}^{n}\lambda_{k}(y^{(i)}_{k})^{2}\bigr)_{i\in\mathbb{N}}$ converges to $\sum_{k=1}^{n}\lambda_{k}\xi_{k}^{2}$. By (0.1) and Step 0, $\sum_{k=1}^{n}\lambda_{k}(y^{(i)}_{k})^{2}\le|y^{(i)}|_{V}^{2}\le R^{2}$ for every $i$, and the constant sequence with value $R^{2}$ converges to $R^{2}$ by Limit of a Sequence of Real Numbers. Claim 1 of Order Properties of Limits of Real Sequences therefore gives $\sum_{k=1}^{n}\lambda_{k}\xi_{k}^{2}\le R^{2}$.

The terms $\lambda_{k}\xi_{k}^{2}$ are nonnegative and the partial sums of $\sum_{k=1}^{\infty}\lambda_{k}\xi_{k}^{2}$ are bounded above by $R^{2}$, so by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion the series converges and its sum, the supremum of the partial sums, is at most $R^{2}$. Hence $x\in V$ by the definition of $V$ in The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights, and by Step 0, $|x|_{V}^{2}=\sum_{k=1}^{\infty}\lambda_{k}\xi_{k}^{2}\le R^{2}$. Since $0\le|x|_{V}$ and $0\le R$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $|x|_{V}\le R$.

Conclusion. The natural numbers $j_{1}<j_{2}<j_{3}<\cdots$ of Step 2 and the point $x\in V$ with $|x|_{V}\le R$ satisfy the requirement that $(x_{j_{i}})_{i\in\mathbb{N}}$ converges to $x$ in $(H,d_{H})$, by Step 5.