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

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, let $\mathbb{N}$ be the set of natural numbers, and let $(H,V,A)$ be a Hilbert triple, with the inner products, norms and distances of $H$ and $V$ written with the subscripts $H$ and $V$, such that $(V,d_{V})$ is separable. Let $C\in\mathbb{R}$ and let $(x_{m})_{m\in\mathbb{N}}$ be a sequence in $V$ with $|x_{m}|_{V}\le C$ for every $m\in\mathbb{N}$. Convergence and weak convergence in $H$ or in $V$ refer to the respective distance and inner product. Then the following hold; since $C$ and $(x_{m})_{m\in\mathbb{N}}$ are arbitrary, each claim holds for every real $C$ and every sequence in $V$ bounded by $C$ in $|\cdot|_{V}$.

1. (Weakly convergent subsequence)¶ There exist a subsequence $(x_{n_{j}})_{j\in\mathbb{N}}$ of $(x_{m})_{m\in\mathbb{N}}$ and a point $w\in V$ with $|w|_{V}\le C$ such that $(x_{n_{j}})_{j\in\mathbb{N}}$ converges weakly to $w$ in $V$ and converges weakly to $w$ in $H$.

2. (Closure)¶ If $(x_{m})_{m\in\mathbb{N}}$ converges to a point $x\in H$ in $(H,d_{H})$, then $x\in V$, $|x|_{V}\le C$, and $(x_{m})_{m\in\mathbb{N}}$ converges weakly to $x$ in $V$.