I need a push with the following theorem, thanks in advance.(adsbygoogle = window.adsbygoogle || []).push({});

Let X and Y be normed spaces, and A : X --> Y a linear operator. A is continuous iff A is bounded.

So, let A be continuous. Then it is continuous at 0, and hence, for [itex]\epsilon = 1[/itex] there exists [itex]\delta > 0[/itex] such that for all x from X such that ||x - 0|| = ||x||<[itex]\delta[/itex], we have ||Ax - A0|| = ||Ax|| < 1. This is where I'm stuck.

For the other direction, let A be bounded. So, there exists some M > 0 such that ||Ax|| [itex]\leq[/itex] M, for all x in X. No further inspiration.

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# Homework Help: Equivalence of continuity and boundedness

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