TUT ANA2 1

Applications of Theorem 314.3

Corollary 330.1(Corollary).

Let $X$ be a Banach space. If a sequence of bounded operators $(T_n)$ converges pointwise, that is, the limit of $(T_n(x))$ exists for all $x\in X$, then these pointwise limits define a bounded linear operator $T$.

Proof.

It is clear that the pointwise limit$T(x)={\lim }_{n\to \infty }{T}_n(x)$ is a linear operator. $\{T_n(x)\}$ is bounded for each $x\in X$. Theorem 314.3 implies $M:={\sup }_{n\in \mathbb{N}}\Vert T_n\Vert <\infty$. Now for any $x\in X$,

$$\begin{array}{r}|T(x)|=\left|\underset{n\to \infty }{\lim }{T}_n(x)\right|=\underset{n\to \infty }{\lim }\left|T_n(x)\right|⩽M|x|,\end{array}$$

Thus, $\Vert T\Vert ⩽M$.□

d4a27f

Proposition 330.2(Proposition).

The convergence $\{T_n\}\to T$ in Corollary 1 is uniform on a compact subset $K\subseteq X$.

Proof.

Let $K\subseteq X$ be compact. Fix $ϵ>0$. Let $M=\text{max}\{{\sup }_{n\in N}\Vert T_n\Vert ,\Vert T\Vert \}$. Cover $K$ by a finite set of open balls $\{B_r(x_i){\}}_{i=1,\dots ,N}$ of radius $r=ϵ/M$. Since $T_n\to T$ pointwise on each of $x_1,\dots ,x_N$, for all large $n$, $\Vert T_n(x_i)-T(x_i)\Vert <ϵ$ for all $i=1,\dots ,N$. By the triangle inequality, we find for all large $n$, for all $x\in K$, $\Vert T_n(x)-T(x)\Vert ⩽3ϵ$.□

[!Example]
Let $\{a_n\}$. If for all $\{x_n\}\in {\ell }^1$, $\sum a_n{x}_n$ converges, then $\{a_n\}\in {\ell }^{\infty }$.

[!Example]

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The diagonal argument

Proposition 330.3(Proposition).

Let $E$ be a countable set. Let $f_n:E\to \mathbb{R}$ be such that $\{f_n(x){\}}_{n=1}^{\infty }$ is bounded for each $x\in E$. Then, there exists a subsequence $\{f_{n_k}{\}}_{k=1}^{\infty }$ such that $\{f_{n_k}(x){\}}_{k=1}^{\infty }$ is convergent for all $x\in E$.