Banach Steinhaus Theorem - Normed space

**dori1123** · March 28th 2009, 01:00 PM

Let $F$ be a collection of continuous linear maps from a Banach space $X$ into a normed space $Y$ . Then the following are equivalent:
(1) $sup_{T \in F}||Tx||< \infty \forall x \in X$
(2) there exists $t>0$ such that $||Tx|| \leq t \forall x \in X$ with $||x|| \leq 1$ and $T \in F$
(3) there exists $t>0$ such that $||Tx|| \leq t||x|| \forall x \in X$ and $T \in F$

**Opalg** · March 29th 2009, 11:56 AM

Originally Posted by dori1123

Let $F$ be a collection of continuous linear maps from a Banach space $X$ into a normed space $Y$ . Then the following are equivalent:
(1) $sup_{T \in F}||Tx||< \infty \forall x \in X$
(2) there exists $t>0$ such that $||Tx|| \leq t \forall x \in X$ with $||x|| \leq 1$ and $T \in F$
(3) there exists $t>0$ such that $||Tx|| \leq t||x|| \forall x \in X$ and $T \in F$

That is the statement of the Banach–Stainhaus theorem, and you can find a proof of it here. Is that what you wanted?

**dori1123** · March 30th 2009, 01:22 PM

I still don't know how to prove (3) ==> (1), can someone explain? Thank you.

**Opalg** · March 30th 2009, 11:37 PM

Originally Posted by dori1123

I still don't know how to prove (3) ==> (1), can someone explain? Thank you.

$(3)\,\Rightarrow\,(1)$ is almost immediate. For a fixed x, $\sup_{T\in F}\|Tx\|\leqslant t\|x\|<\infty$ .

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