Equivalence of Boundedness and Continuity
As stated in the introduction, a linear operator L between normed spaces X and Y is bounded if and only if it is a continuous linear operator. The proof is as follows.
- Suppose that L is bounded. Then, for all vectors v and h in X with h nonzero we have
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- Letting go to zero shows that L is continuous at v. Moreover, since the constant M does not depend on v, this shows that in fact L is uniformly continuous (Even stronger, it is Lipschitz continuous.)
- Conversely, it follows from the continuity at the zero vector that there exists a such that for all vectors h in X with . Thus, for all non-zero in X, one has
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- This proves that L is bounded.
Read more about this topic: Bounded Operator
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