I have some questions about proof of the theorem below is written in my notes. If someone help me about it I will be appreciated. Thanks
Theorem : Let $(X,\|.\|_X)$ is a normed space and $(Y,\|.\|_Y)$ is a Banach space then $(B(X,Y),\|.\|_{op})$ is a Banach space
I have problem about some writtens :
1) $\|T_n(x)-T_m(x)\|_Y \leq \|T_n-T_m\|_{op}$
$\|T_n(x)-T_m(x)\|_Y \leq sup_{\|x\|_X \leq 1}\|T_n(x)-T_m(x)\|_Y$ it is true only for some $x \in X$ not for all. How can we use it?
2) $T_n(x) \to y , \exists y \in Y$ since $Y$ is a Banach space.It's OK for me but after this statement "Since $\forall x \in X$ $\exists y \in Y$ we can write $lim_{n\to \infty}T_n(x)=T(x)$" is written.
How can we write it?
Thanks in advance :)
For the first point, in general you don't have $\|T_n(x)-T_m(x)\|_Y \leq \|T_n-T_m\|$. Rather we have $\|T_nx - T_mx\| \leq \|T_n - T_m\| \cdot \|x\|$.
This follows immediately from noting that for $\|x\| \leq 1$, $$\|T_n(x)-T_m(x)\|_Y \leq sup_{\|x\|_X \leq 1}\|T_n(x)-T_m(x)\|_Y$$ (consider $\frac{x}{\|x\|}$). This is enough to show that if $(T_n)$ is Cauchy in $B(X,Y)$ then for every fixed $x$, $(T_nx)$ is Cauchy in $Y$ which is what you want.
For the second point, all that they mean is that one can define a map $T:X \to Y$ by setting $Tx = \lim_\limits{n\to \infty} T_nx$. They aren't claiming that this $T$ is then surjective - notice that the statement is instead that for every $x$ we can define $Tx$. In general, $T$ need not be surjective. It is however always a bounded linear map which is what you want.