Proof of Weak Compactness and Closedness of Bounded Subsets of the Small Space of a Hilbert Triple

We use Bounded Sequences in a Separable Real Hilbert Space Have Weakly Convergent Subsequences, Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator, Elementary Properties of Weak Convergence in a Real Inner Product Space and The Subsequence Criterion for Convergence in a Metric Space.

Claim 1. $V$ is a real Hilbert space with $(V,d_{V})$ separable, so Bounded Sequences in a Separable Real Hilbert Space Have Weakly Convergent Subsequences applied in $V$ gives a subsequence $(x_{n_{j}})_{j}$ of $(x_{m})$ and $w\in V$ with $|w|_{V}\le C$ such that $x_{n_{j}}\rightharpoonup w$ in $V$. By Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator (ii) the subsequence also converges weakly to $w$ in $H$.

Claim 2. Suppose $(x_{m})$ converges to $x$ in $(H,d_{H})$. Let $(x_{n_{j}})_{j}$ be any subsequence. By claim 1 applied to this subsequence (which is still bounded by $C$ in $V$), there are a further subsequence $(u_{i})_{i}=(x_{n_{j_{i}}})_{i}$ and $w\in V$ with $|w|_{V}\le C$ such that $u_{i}\rightharpoonup w$ in $V$ and in $H$. Since $(u_{i})$ is a subsequence of $(x_{m})$ (The Subsequence Criterion for Convergence in a Metric Space (i)), it converges to $x$ in $H$ by A Subsequence of a Convergent Sequence Has the Same Limit, hence weakly to $x$ in $H$ by Elementary Properties of Weak Convergence in a Real Inner Product Space (ii); by Elementary Properties of Weak Convergence in a Real Inner Product Space (i) in $H$, $w=x$. Applying this to $(x_{m})$ itself, the subsequence determined by the identity index sequence $n_{j}=j$ (strictly increasing by claim 5 of Properties of the Order on the Natural Numbers), shows $x=w\in V$ and $|x|_{V}\le C$.

Finally, $(x_{m})$ converges weakly to $x$ in $V$: fix $y\in V$ and consider the real sequence $(\langle x_{m},y\rangle_{V})_{m}$ in the metric space $(\mathbb{R},d_{\mathbb{R}})$ of The Absolute Value Metric on the Real Numbers. By what was shown, every subsequence $(x_{n_{j}})$ has a further subsequence $(u_{i})$ with $u_{i}\rightharpoonup x$ in $V$, so that $(\langle u_{i},y\rangle_{V})_{i}$ converges to $\langle x,y\rangle_{V}$ (Weak Convergence of a Sequence in a Real Inner Product Space, read in $(\mathbb{R},d_{\mathbb{R}})$ via claim 1 of Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete). Hence every subsequence of $(\langle x_{m},y\rangle_{V})_{m}$ has a further subsequence converging to $\langle x,y\rangle_{V}$, and The Subsequence Criterion for Convergence in a Metric Space (ii) gives that $(\langle x_{m},y\rangle_{V})_{m}$ converges to $\langle x,y\rangle_{V}$. As $y$ was arbitrary, $x_{m}\rightharpoonup x$ in $V$.