Let f: H H be a bounded linear functional on a separable Hilbert space H (with...

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Let f: H→ H be a bounded linear functional on a separable Hilbert space H (with inner product denoted by (+,-)). Prove that there is a unique element y E H such that f(x) = (x,y) for all Hand ||f|| = ||y||. Hint. You may use the following facts: A separable Hilbert space, H, contains a complete orthonormal sequence, {}, satisfying the following properties: (1) If r, y EH and if (r, ok) = (y, ox) for all k, then x = y. (2) Parseval's equality holds; that is, for all z € H. (x,x) = a., where a = (x,x). k=1 Let f: H→ H be a bounded linear functional on a separable Hilbert space H (with inner product denoted by (+,-)). Prove that there is a unique element y E H such that f(x) = (x,y) for all Hand ||f|| = ||y||. Hint. You may use the following facts: A separable Hilbert space, H, contains a complete orthonormal sequence, {}, satisfying the following properties: (1) If r, y EH and if (r, ok) = (y, ox) for all k, then x = y. (2) Parseval's equality holds; that is, for all z € H. (x,x) = a., where a = (x,x). k=1

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