# Series of implication for a sequence of bounded linear transformations

Let X be a normed space and Y a Banach space, and let $$\{T_n\}_{n \in \mathbb{N}} \subset B(X,Y)$$, show that the following are equivalent:

i) $$sup_{n \in \mathbb{N}} \|T_{N}\| < +\infty$$;

ii) for any $$x \in X$$, $$sup_{n \in \mathbb{N}} \|T_n(x)\| <+\infty$$;

iii) for any $$\phi \in Y^*$$, $$sup_{n \in \mathbb{N}}|\phi(T_n(x))| < +\infty$$.

Here $$Y^*$$ denotes the topological dual of the space

My attempt so far:

For i -> ii I argued that for any $$x \in X \; \|T_n(x)\| < \|T_n\|\|x\|$$

As for ii -> iii I used a similar argument as before by saying that the norm of $$\phi$$ is finite

As for the rest, my teacher advised we prove that iii->ii and ii->i. Now ii->i is pretty simple, it's the well known Banach-Steinhaus theorem, but I don't see a way to prove iii -> ii.

• iii) says "for any $\phi$" but it doesn't say "for any $x$". Jul 2, 2021 at 0:44
The proof of $$iii\rightarrow ii$$ is also an application of Banach Steinhaus. Let $$sup_{n \in \mathbb{N}}|\phi(T_n(x))| < +\infty\Rightarrow sup_{n \in \mathbb{N}} |(T_n(x))^{**}(\phi)|<\infty$$ where $$(T_n(x))^{**}:X^*\rightarrow \mathbb{C}$$ s.t. $$(T_n(x))(\phi)=\phi(T_n(x))$$ and the map $$X\rightarrow X^{**}$$ where $$x\mapsto x^{**}$$ is an isometric embedding i.e. $$\|T_n(x)\|=\|T_n(x)^{**}\|$$. Now applying Banach Steinhaus on the space $$X^*$$, you get that $$sup_{n \in \mathbb{N}} \|{(T_n(x))^{**}}\|<\infty\Rightarrow sup_{n \in \mathbb{N}} \|(T_n(x))\|<\infty$$.
Little more details: Define $$X^{**}$$ to be the dual space of $$X^*$$ i.e. $$\{\psi:X^*\rightarrow \mathbb{C} \ni \psi$$ is a bounded linear functional$$\}$$. Now there is an embedding of $$X\rightarrow X^{**}$$ where $$x\mapsto x^{**}$$ such that $$x^{**}(\phi)=\phi(x)$$. You can easily show that $$x^{**}$$ is a bounded linear functional by a Corollary of the Hahn Banach theorem. The corollary states that given $$x\in X$$ $$\exists \phi_x\in X^* \ni \phi_x(x)=\|x\|$$ and $$\|\phi_x\|=1$$.
First oberve that $$\|x^{**}\|=sup_{\phi\in X^*(\neq 0)}\frac{|x^{**}(\phi)|}{\|\phi\|}=sup_{\phi\in X^*(\neq 0)}\frac{|\phi(x)|}{\|\phi\|}\leq sup_{\phi\in X^*(\neq 0)}\frac{\|\phi\|\|x\|}{\|\phi\|} =\|x\|$$. Thus $$\|x^{**}\|\leq \|x\|$$. Again by the corollary stated above $$\exists \phi_x\in X^* \ni \phi_x(x)=\|x\|$$ and $$\|\phi_x\|=1$$. This implies $$x^{**}(\phi_x)=\phi_x(x)=\|x\|$$, and hence $$\|x^{**}\|= \|x\|$$. Thus $$x^{**}$$ is a bounded linear functional.