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**kinkong** Question:

Let (ej) be an orthonormal sequence in Hilbert space H. Show that if x=\sum_{j=1}^\infty {\alpha}_{j}{e}_{j}

y= \sum_{j=1}^\infty {\\beta }_{j}{e}_{j} then <x,y>=\sum_{j=1}^\infty {\alpha}_{j}{\bar{\beta}}_{j}, the series being absolutely convergent.

solution:

x=\sum_{j=1}^\infty {\alpha}_{j}{e}_{j} ==> ||x|| = \sum_{j = 1}^\infty {|{\alpha}_{j}|}^{2} ....(1)

y= \sum_{j=1}^\infty {\\beta }_{j}{e}_{j}==>||y|| = \sum_{j = 1}^\infty {|{\beta }_{j}|}^{2} ....(2)

I need to show that {S}_{ n} = \sum_{j=1}^\infty {\alpha}_{j}{\bar{\beta}}_{j} converges and the limit is <x,y>

i tried to define

{x}_{n} =\sum_{j=1}^\infty {\alpha}_{j}{e}_{j} ==> ||x|| = \sum_{j = 1}^\infty {|{\alpha}_{j}|}^{2}

{y}_{n } = \sum_{j=1}^\infty {\\beta }_{j}{e}_{j}==>||y|| = \sum_{j = 1}^\infty {|{\beta }_{j}|}^{2}

so that i can use (1) and 92) to show that <{x}_{n},{y}_{n }> \to <x,y> as n\to \infty

i couldnt go from here.

Please can someone help me