Proof of Elementary Properties of Weak Convergence in a Real Inner Product Space

We use the identities of Elementary Identities in a Real Inner Product Space, the Cauchy-Schwarz inequality The Cauchy-Schwarz Inequality in a Real Inner Product Space, the continuity statement The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, and for real sequences Arithmetic of Limits of Real Sequences, Order Properties of Limits of Real Sequences and Uniqueness of Limits and Boundedness of Convergent Real Sequences. Real-number facts are taken from Elementary Order Arithmetic in an Ordered Field, Elementary Arithmetic in an Ordered Field and Properties of the Absolute Value in an Ordered Field. A constant real sequence converges to its value (Constant Sequences and Index-Shifted Sequences of Real Numbers §constant).

Claim 1. For every $z\in E$ the real sequence $(\langle x_{m},z\rangle)$ converges both to $\langle x,z\rangle$ and to $\langle y,z\rangle$, so $\langle x,z\rangle=\langle y,z\rangle$ by claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences, that is, $\langle x-y,z\rangle=0$ (Elementary Identities in a Real Inner Product Space §bilinear). With $z=x-y$ this gives $|x-y|^{2}=0$, so $x=y$ by Elementary Identities in a Real Inner Product Space §vanishing.

Claim 2. By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, for every $z\in E$ the sequence $(\langle x_{m},z\rangle)$ converges to $\langle x,z\rangle$.

Claim 3. For $z\in E$, $\langle x_{m}\pm y_{m},z\rangle=\langle x_{m},z\rangle\pm\langle y_{m},z\rangle$ and $\langle\lambda x_{m},z\rangle=\lambda\langle x_{m},z\rangle$, and the right sides converge to $\langle x,z\rangle\pm\langle y,z\rangle=\langle x\pm y,z\rangle$ and $\lambda\langle x,z\rangle=\langle\lambda x,z\rangle$ by claims 1 and 3 of Arithmetic of Limits of Real Sequences.

Claim 4. Let $(x_{n_{k}})$ be a subsequence and $z\in E$. The real sequence $(\langle x_{n_{k}},z\rangle)_{k}$ is the subsequence of $(\langle x_{m},z\rangle)_{m}$ determined by $(n_{k})$; by claim 1 of Convergence and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space the latter converges to $\langle x,z\rangle$ in $(\mathbb{R},d_{\mathbb{R}})$, hence so does the subsequence by A Subsequence of a Convergent Sequence Has the Same Limit, and the same claim 1 turns this back into convergence of real numbers.

Claim 5. First, $0\le|x_{1}|\le C$, so $0\le C$. By Cauchy-Schwarz and claim 5 of Elementary Arithmetic in an Ordered Field, $\langle x_{m},x\rangle\le|\langle x_{m},x\rangle|\le|x_{m}|\,|x|\le C|x|$ for every $m$ (claim 3 of Properties of the Absolute Value in an Ordered Field for the first step). The left side converges to $\langle x,x\rangle=|x|^{2}$, so $|x|^{2}\le C|x|$ by claim 1 of Order Properties of Limits of Real Sequences applied against the constant sequence $(C|x|)$. If $|x|=0$ then $|x|\le C$ holds as $0\le C$. Otherwise $|x|>0$ and multiplying by $|x|^{-1}$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $|x|\le C$.

Claim 6. By Elementary Identities in a Real Inner Product Space §expansion, $|x_{m}-x|^{2}=|x_{m}|^{2}-2\langle x_{m},x\rangle+|x|^{2}$. The right side converges to $|x|^{2}-2|x|^{2}+|x|^{2}=0$ by claims 1, 2 and 3 of Arithmetic of Limits of Real Sequences (the sequence $(|x_{m}|^{2})$ being the product of $(|x_{m}|)$ with itself). Given $\varepsilon>0$, choose $N$ with $|x_{m}-x|^{2}<\varepsilon^{2}$ for $m\ge N$ (using $\varepsilon^{2}>0$, claim 5 of Elementary Order Arithmetic in an Ordered Field); then $|x_{m}-x|<\varepsilon$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Hence $(x_{m})$ converges to $x$ in $(E,d)$.

Claim 7. As in claim 5, $0\le|x_{1}|\le C$, so $0\le C$ and $0<C+1$. By Elementary Identities in a Real Inner Product Space §bilinear, claim 5 of Properties of the Absolute Value in an Ordered Field and Cauchy-Schwarz,

Given $\varepsilon>0$, choose $N_{1}$ with $|y_{m}-y|<\varepsilon/(2(C+1))$ for $m\ge N_{1}$ (possible as $(y_{m})$ converges to $y$ and $C+1>0$) and $N_{2}$ with $|\langle x_{m},y\rangle-\langle x,y\rangle|<\varepsilon/2$ for $m\ge N_{2}$; for $m\ge\max(N_{1},N_{2})$ (claim 1 of Elementary Properties of the Maximum of Two Elements) the right side is less than $\varepsilon$, using $C\le C+1$.

Claim 8. As in claim 5, $0\le C$. Let $y\in E$ and $\varepsilon>0$, and put $K=C+|x|+1>0$. Since $y$ lies in the closure of $D$, Sequential Characterization of the Closure in a Metric Space gives a sequence in $D$ converging to $y$, hence some $z\in D$ with $|y-z|<\varepsilon/(3K)$. For every $m$, by bilinearity, the triangle inequality for real numbers and Cauchy-Schwarz,

The first and third terms are each less than $\varepsilon/3$ (since $C\le K$ and $|x|\le K$), and by hypothesis there is $N$ with the middle term less than $\varepsilon/3$ for $m\ge N$. Hence $|\langle x_{m},y\rangle-\langle x,y\rangle|<\varepsilon$ for $m\ge N$, so $(\langle x_{m},y\rangle)$ converges to $\langle x,y\rangle$; as $y$ was arbitrary, $x_{m}\rightharpoonup x$.