What will be the next step to ensure that the pointwise convergence implies convergence in the operator norm?

$$(X, \|•\|)$$and $$(Y, \|•\|')$$ be two normed space and \begin{align} {\scr{B}}{(X, Y) }&=\{T\in {\scr{L}}{(X,Y)} : T \text { is bounded } \}\end{align}

$$\|T\|_{op}=\sup_{\|x\|\le1}\|Tx\|'$$

I want to understand the convergence of sequences in the space $${(\scr{B}}{(X, Y) }, \|•\|_{op})$$

$$T_n\to T$$ pointwise on $$X$$ i.e $$\forall x\in X, T_n x \to Tx \text { in the space }\space \space (Y, \|•\|')$$.

Question : Does this implies $$T_n\to T$$ in the space $${(\scr{B}}{(X, Y) }, \|•\|_{op})$$?

1. $$T_n\to T$$ pointwise implies $$T\in {\scr{L}}{(X, Y) }$$

Proof: $$\forall$$ $$x, y\in X$$ and $$\lambda \in \mathbb{K}$$

\begin{align}T(x+\lambda y) &=\lim_{n\to\infty} T_n(x+\lambda y)\\&=\lim_{n\to\infty} T_nx +\lambda T_ny\\&=\lim_{n\to \infty} T_nx+\lambda \lim_{n\to\infty}T_ny\\&=Tx+\lambda Ty \end{align}

1. $$T$$ may not be continuous.

For an example, choose $$(\ell_1, \|•\|_{\infty})$$ and

$$(f_n) \in (\ell_1,\|•\|_{\infty})^{*}$$ defined by $$f_n(x) =\sum_{k=1}^{n} x_k$$

Where $$x=(x_1, x_2,...) \in\ell_1$$

Then, $$f_n\to f$$ pointwise where$$f(x) =\sum_{k\in \Bbb{N}} x_k$$

But $$f\notin (\ell_1,\|•\|_{\infty})^{*}$$

Here we need $$(X, \|•\|)$$ to be a Banach space to ensure that $$T\in {\scr{B}}{(X, Y) }$$ [Banach-Steinhaus Theorem]

1. Suppose in addition, $$(X, \|•\|)$$ is a Banach space and then $$T\in {\scr{B}}{(X, Y) }$$ but still $$(T_n)$$ may not converge to $$T$$ in the space $${(\scr{B}}{(X, Y) }, \|•\|_{op})$$

For an example, choose $$(c_0, \|•\|_{\infty})$$ and $$(f_n)$$ defined by $$f_n(x) =x_n$$ where $$x=(x_1, x_2,...) \in c_0$$ i.e converges to $$0$$.

Then, $$f_n\to 0$$ pointwise but $$f_n \nrightarrow 0$$ in the Dual space.

My question : What will be the next step to ensure that the pointwise convergence implies convergence in the operator norm?

• There is no useful sufficient condition for that. Apr 27 at 8:24

As Kavi mentioned, there are no good answers to your question. Pointwise convergence is strictly weaker than convergence in $$\|\cdot\|_{op}$$, so you will need to add fairly strong hypotheses to the former in order to get the latter.
1. The domain $$X$$ is finite-dimensional.
2. All of the $$T_n$$ lie in some fixed finite-dimensional subspace of $${\scr B}(X, Y)$$.
3. $$\{T_n\}_n$$ is a Cauchy sequence for $$\|\cdot\|_{op}$$.
4. When verifying that $$T_n(x)→T(x)$$, that is, that for every $$ε>0$$ there is some $$n_0$$ such that $$n≥n_0 \Rightarrow \|T_n(x)-T(x)\|<\epsilon$$, you are able to exhibit $$n_0$$ depending only on $$\epsilon$$ and $$\|x\|$$ (rather than $$\epsilon$$ and $$x$$).