Sequence of bounded linear operators implicating Cauchy sequence in $\mathbb K$

Question

Let H be a Hilbert space and $(T_n)_{n \in \mathbb N}$ be a sequence in ${\rm BL}(H)$ (bounded linear operators) such that $(\langle y,T_nx \rangle)_{n \in \mathbb N}$ is a Cauchy sequence in $\mathbb K$ for all $x,y \in H$. Prove that there exists $T \in {\rm BL}(H)$ such that $$T_n \xrightarrow{w} T$$ as $n \rightarrow \infty$ (weak convergence).

It carries on about general equivalences and the norm in weak convergence but showing that seems quite straightforward yet I'm pretty clueless on how to tackle this particular problem. So I'd appreciate any help.

Answer 1

First of all, your requirements imply that for each $x \in H$, the sequence $\iota(T_n x)$ of bounded linear functionals, where $\iota$ denotes the canonical embedding of $H$ into the dual $H'$ by $\iota(x)(y) := \langle y, x\rangle$ is pointwise bounded.

By the uniform boundedness principle, this implies that $\Vert T_n x \Vert$ is a bounded sequence for every $x \in H$. Using the uniform boundedness principle again, we conclude that $\Vert T_n \Vert$ is a bounded sequence, say $\Vert T_x \Vert \leq R$ for all $n$.

Now look at the answer by Robert Israel, or see below

By completeness of $\Bbb{K}$, you know that the limit $$f_{x,y} := \lim_n \langle y, T_n x \rangle$$ exists for all $x,y \in H$ and by the above, we know that \begin{equation*} |f_{x,y}| \leq \Vert y \Vert \cdot \Vert T_n x \Vert \leq R \vert y \Vert \cdot \Vert x \Vert. \end{equation*} It is easy to see that $y \mapsto f_{x,y}$ is linear for each $x \in H$. By the Riesz representation theorem, there is thus a unique $z_x \in H$ with $\Vert z_x \Vert \leq >!R \Vert x \Vert$ and \begin{equation*}f_{x,y} = \langle y, z_x \rangle\end{equation*} for all $x \in H$. Now show that $x \mapsto z_x$ is linear (use uniqueness) and conclude that $T : x \mapsto z_x$ is a bounded linear map with \begin{equation*}T_n \xrightarrow{w} T.\end{equation*}

Answer 2

Hint: Let $L(x,y) = \lim_{n \to \infty} \langle y, T_n x \rangle$. You want to define $T$ so that $\langle y, T x \rangle = L(x,y)$. The Principle of Uniform Boundedness will be useful.