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Thread: Orthonormal Sequence in a Hilbert space

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by kinkong View Post
    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
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  3. #3
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    Apr 2011
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    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
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  4. #4
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    México
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    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

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