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Math Help - Dimension of a Hilbert space

  1. #1
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    Dimension of a Hilbert space

    Why is the dimension of the Hilbert space of square-integrable functions infinite? I guess I'm having trouble visualizing the basis of this space.
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    Quote Originally Posted by tjkubo View Post
    Why is the dimension of the Hilbert space of square-integrable functions infinite? I guess I'm having trouble visualizing the basis of this space.
    Why not just show a concrete infinite set of linearly-independent square-integrable functions? (Although, come to think of it, we must know something more about the domain of those square-integrable functions. If that domain can be any measure space, then the proposition might not even be true in general.)
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  3. #3
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    For example, if are talking about the space of square integrable functions on a finite interval, L^2([a, b]) then all functions of theform x^n for positive integer n are in the space. Can you show that they are independent?
    A set of functions like sin(nx) for integer n would be a natural choice also.
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