A linear function f: X Y is continuous if and only if it is continuous at
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A linear function f: X → Y is bounded if there exists a constant M such that
|| f(x) || ≤ M ||x|| for every x ∊ X
Note that boundedness does not imply that the range f (X) is bounded but rather that f (S) is bounded for every bounded set S. As the following exercise demonstrates, boundedness is equivalent to continuity for linear functions. Consequently these two terms are used interchangeably in practice.
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Clearly if is continuous is continuous at 0 To show the converse a...View the full answer
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