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Math Help - Banach Steinhaus Theorem - Normed space

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    Banach Steinhaus Theorem - Normed space

    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
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    Quote Originally Posted by dori1123 View Post
    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?
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    I still don't know how to prove (3) ==> (1), can someone explain? Thank you.
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    Quote Originally Posted by dori1123 View Post
    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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