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How do I show that a linear function from a Hilbert space $H$ to itself is continuous if $H$ is finite dimensional?

Also, what would be an example of a linear function from a Hilbert space to itself which is not continuous?

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    Do you know that a linear operator between Banach spaces is continuous if and only if it is bounded? (You should do this as an exercise first if not.) Can you show that a linear operator from a finite-dimensional Hilbert space to another one must be bounded? Can you write down a linear operator on an infinite-dimensional Hilbert space which isn't bounded?2011-09-15
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    An aside: An example of a discontinuous linear operator on a Hilbert space requires some choice: http://mathoverflow.net/questions/5303/basis-of-linfinity/5313#53132011-09-15

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