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How to interpret the infinity of Hilbert Space?

  1. Mar 20, 2017 #1
    This is basically just a comprehension question, but what makes elements of the Hilbert space exist in infinite dimensions? I understand that the number of base vectors to write out an element, like a wavefunction, are infinite:
    \begin{equation*}
    \psi(x) = \int c_s u_s (x) ds = \sum_k^{\infty} \hat{c}_k \hat{u}_k(x)
    \end{equation*}
    So what are the bases u(x)? Are they just other wavefunctions that build a new wavefunction via superposition? And if so, how does this justify the infinite dimensions of the Hilbert space and why exactly is an infinite number of sub-wavefunctions necessary?

    Also apologies if I posted this in the wrong subforum, not really sure what this question classifies as.

    Thanks in advance.
     
  2. jcsd
  3. Mar 20, 2017 #2

    PeroK

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    Even if you limited yourself to, say, all polynomial functions, then you have an infinite number of basis functions: ##1, x, x^2, \dots##
     
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