Let be an orthonormal set of a Hilbert space. Let and be two vectors in H....

Question

Question

Let $\left \{ e_{k}:k\in \mathbb{N} \right \}$ be an orthonormal set of a Hilbert space. Let

$f = \sum_{k=0}^{\infty} x_{k}e_{k}$ and $g = \sum_{k=0}^{\infty} y_{k}e_{k}$ be two vectors in H. Show that

$\sum_{k=0}^{\infty} x_{k}\overline{y_{k}}$

converges absolutely, and that

$\left \langle f,g\right \rangle=\sum_{k=0}^{\infty} x_{k}\overline{y_{k}}$

Answer 1

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