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Let $X$ and $Y$ be Banah spaces and let $(T_n)$ be a sequence of bounded linear operator from $X$ to $Y$. I need to prove that following statements are equivalent:

(a) Sequence $(||T_n||)$ is bounded

(b) Sequence $(||T_n(x)||)$ is bounded for each $x\in X$

(c) Sequence $(|f(T_n(x))|)$ is bounded for each $x\in X$ and for each linear functionals define on $Y$.

Laki888
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    Obviously (a) $\Rightarrow$ (b) $\Rightarrow$ (c). (b) $\Rightarrow$ (a) follows from the uniform boundedness theorem applied to $(T_n)$, (c) $\Rightarrow$ (b) from the uniform boundedness theorem applied to $J_Y(T_n(x))$, where $J_Y \colon Y \to Y^{**}$ is the natural injection. – martini Dec 10 '13 at 13:44

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Hint: check the Uniform Boundedness Principle in any Functional Analysis textbook.

Yurii Savchuk
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