Originally Posted by
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