1 show that for any two normed linear spaces x imiddoti, Ask an Expert

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1. Show that for any two normed linear spaces (X, I·I) and (Y, |·|), and L(X, Y ) the set of all bounded linear operators from X into Y , the operator norm T f→ IT I is in fact a norm on L(X, Y ).

2. Prove in detail that the operator T with T (en ) = en/n, where {en } is an orthonormal basis of a Hilbert space, is continuous but not open and not onto. Show that its range is dense.

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