Basis and vectors in the Hilbert Space of functions of type $f : mathbbR rightarrow mathbbR$

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Let us take the set of functions, $F = f $. We can model this as a Hilbert space.



Question 1: Can we easily find a basis for this Hilbert space?



Let us suppose we have a basis for this Hilbert space called $B = k in mathbbR $.



Question 2: Can I write a general vector in this space as follows:



$$g(x) = int^infty_-infty(a_1(k)b_1(kx)+ a_2(k)b_2(kx) + ldots )dk$$



Question 3: Under what conditions do vectors of the following form exist?



$$g(x) = sum^infty_i a_i(k_i)b_i(k_i x)$$




functional-analysis




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    What inner product are you using?
    – Ross Millikan
    Aug 8 at 20:31






  • 2




    A Hilbert space is a vector space that also has an inner product and is complete. The same issue with a basis as in your other question applies. Only finite sums of basis vectors are permitted, so showing an explicit basis does not work.
    – Ross Millikan
    Aug 8 at 20:38














up vote
-2
down vote

favorite












Let us take the set of functions, $F = f $. We can model this as a Hilbert space.



Question 1: Can we easily find a basis for this Hilbert space?



Let us suppose we have a basis for this Hilbert space called $B = k in mathbbR $.



Question 2: Can I write a general vector in this space as follows:



$$g(x) = int^infty_-infty(a_1(k)b_1(kx)+ a_2(k)b_2(kx) + ldots )dk$$



Question 3: Under what conditions do vectors of the following form exist?



$$g(x) = sum^infty_i a_i(k_i)b_i(k_i x)$$




functional-analysis




share|cite|improve this question


















  • 4




    What inner product are you using?
    – Ross Millikan
    Aug 8 at 20:31






  • 2




    A Hilbert space is a vector space that also has an inner product and is complete. The same issue with a basis as in your other question applies. Only finite sums of basis vectors are permitted, so showing an explicit basis does not work.
    – Ross Millikan
    Aug 8 at 20:38












up vote
-2
down vote

favorite









up vote
-2
down vote

favorite











Let us take the set of functions, $F = f $. We can model this as a Hilbert space.



Question 1: Can we easily find a basis for this Hilbert space?



Let us suppose we have a basis for this Hilbert space called $B = k in mathbbR $.



Question 2: Can I write a general vector in this space as follows:



$$g(x) = int^infty_-infty(a_1(k)b_1(kx)+ a_2(k)b_2(kx) + ldots )dk$$



Question 3: Under what conditions do vectors of the following form exist?



$$g(x) = sum^infty_i a_i(k_i)b_i(k_i x)$$




functional-analysis




share|cite|improve this question














Let us take the set of functions, $F = f $. We can model this as a Hilbert space.



Question 1: Can we easily find a basis for this Hilbert space?



Let us suppose we have a basis for this Hilbert space called $B = k in mathbbR $.



Question 2: Can I write a general vector in this space as follows:



$$g(x) = int^infty_-infty(a_1(k)b_1(kx)+ a_2(k)b_2(kx) + ldots )dk$$



Question 3: Under what conditions do vectors of the following form exist?



$$g(x) = sum^infty_i a_i(k_i)b_i(k_i x)$$





functional-analysis





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 8 at 20:38

























asked Aug 8 at 20:18









Ben Sprott

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  • 4




    What inner product are you using?
    – Ross Millikan
    Aug 8 at 20:31






  • 2




    A Hilbert space is a vector space that also has an inner product and is complete. The same issue with a basis as in your other question applies. Only finite sums of basis vectors are permitted, so showing an explicit basis does not work.
    – Ross Millikan
    Aug 8 at 20:38












  • 4




    What inner product are you using?
    – Ross Millikan
    Aug 8 at 20:31






  • 2




    A Hilbert space is a vector space that also has an inner product and is complete. The same issue with a basis as in your other question applies. Only finite sums of basis vectors are permitted, so showing an explicit basis does not work.
    – Ross Millikan
    Aug 8 at 20:38







4




4




What inner product are you using?
– Ross Millikan
Aug 8 at 20:31




What inner product are you using?
– Ross Millikan
Aug 8 at 20:31




2




2




A Hilbert space is a vector space that also has an inner product and is complete. The same issue with a basis as in your other question applies. Only finite sums of basis vectors are permitted, so showing an explicit basis does not work.
– Ross Millikan
Aug 8 at 20:38




A Hilbert space is a vector space that also has an inner product and is complete. The same issue with a basis as in your other question applies. Only finite sums of basis vectors are permitted, so showing an explicit basis does not work.
– Ross Millikan
Aug 8 at 20:38















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