# Thread: Orthonormal Sequence in a Hilbert space

1. ## Orthonormal Sequence in a Hilbert space

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

2. 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

Go to the LaTeX section help in MHF . Your message is almost impossible to read with all those awful symbols.

Tonio

3. Question:
Let (ej) be an orthonormal sequence in Hilbert space H. Show that if $\displaystyle x=\sum_{j=1}^\infty {\alpha}_{j}{e}_{j}$
$\displaystyle y= \sum_{j=1}^\infty {\beta }_{j}{e}_{j}$ then $\displaystyle <x,y>=\sum_{j=1}^\infty {\alpha}_{j}{\bar{\beta}}_{j}$ , the series being absolutely convergent.

solution:
$\displaystyle x=\sum_{j=1}^\infty {\alpha}_{j}{e}_{j} ==> ||x|| = \sum_{j = 1}^\infty {|{\alpha}_{j}|}^{2} ....(1)$
$\displaystyle y= \sum_{j=1}^\infty {\beta }_{j}{e}_{j}==>||y|| = \sum_{j = 1}^\infty {|{\beta }_{j}|}^{2} ....(2)$

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

i tried to define
$\displaystyle {x}_{n} =\sum_{j=1}^\infty {\alpha}_{j}{e}_{j} ==> ||x|| = \sum_{j = 1}^\infty {|{\alpha}_{j}|}^{2}$
$\displaystyle {y}_{n } = \sum_{j=1}^\infty {\beta }_{j}{e}_{j}==>||y|| = \sum_{j = 1}^\infty {|{\beta }_{j}|}^{2}$

so that i can use (1) and (2) to show that $\displaystyle <{x}_{n},{y}_{n }> \to <x,y>$ as n\to \infty

4. This is just expressing the fact that the inner product is continous in the product space. To prove this assume $\displaystyle x_n \rightarrow x$ and $\displaystyle y_n \rightarrow y$ in norm then

$\displaystyle 0\leq |\langle x_n,y_n \rangle - \langle x,y\rangle | \leq |\langle x_n-x,y_n\rangle |+|\langle x,y_n-y\rangle | \leq \|y_n \| \| x_n-x\| + \| x\| \| y_n-y\| \rightarrow 0$