Prove that the norm, , considered as a function is continuous. i.e. , converges wrt the norm then,

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- January 26th 2009, 05:39 PMPurProve that the norm on L^2 space is continuous
Prove that the norm, , considered as a function is continuous. i.e. , converges wrt the norm then,

- January 27th 2009, 12:16 AMOpalg
This is true in any normed space, and is a simple consequence of the triangle inequality. Start with the fact that , so that . The same reasoning with f and f_n interchanged then gives .