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If $X$ and $Y$ are normed spaces I know that the space $B(X,Y)$ of bounded linear functions from $X$ to $Y$, is complete if $Y$ is complete. Is there an example of a pair of normed spaces $X,Y$ s.t. $B(X,Y)$ is not complete?

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up vote 3 down vote accepted

Let $X = \mathbb{R}$ with the Euclidean norm and let $Y$ be a normed space which is not complete. You should find that $B(X, Y) \simeq Y$.

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Yeah I see it. for $f \in B(X,Y)$, $f(1)$ completly determines $f$, so $B(X,Y) \simeq Y$. Thanks. – Digital Gal Aug 6 '10 at 1:00

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