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Math Help - Every vector space over R or C can be made into an inner product space

  1. #1
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    Every vector space over R or C can be made into an inner product space

    Let V be a vector space over F = R or C.

    For the case when dim V over F is finite, say n, if we let B be a basis for V,
    a vector v in V has a corresponding coordinate vector with respect to the ordered basis B. Denote this representation as [v] = (a_1 ... a_n).

    It is not difficult to check that
    f(v,w) = sum of (conjugate(a_i) times b_i) where [v] = (a_1 ... a_n)
    and [w]=(b_1 ... b_n) will work as an inner product.
    Hence, V is an inner product space over F.

    But I don't have an idea on what inner product to consider for the case when V is infinite dimensional over F= R or C.
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  2. #2
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    Quote Originally Posted by mysteriouspoet3000 View Post
    Let V be a vector space over F = R or C.

    For the case when dim V over F is finite, say n, if we let B be a basis for V,
    a vector v in V has a corresponding coordinate vector with respect to the ordered basis B. Denote this representation as [v] = (a_1 ... a_n).

    It is not difficult to check that
    f(v,w) = sum of (conjugate(a_i) times b_i) where [v] = (a_1 ... a_n)
    and [w]=(b_1 ... b_n) will work as an inner product.
    Hence, V is an inner product space over F.

    But I don't have an idea on what inner product to consider for the case when V is infinite dimensional over F= R or C.


    Just generalize the finite dimensional case: choose a basis for the vector space (AC is needed here for

    inf. dim. cases) and define the inner product as before, taking into account that any element in the v.s. is

    a FINITE lin. comb. of some elements in the basis.

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    Just generalize the finite dimensional case: choose a basis for the vector space (AC is needed here for

    inf. dim. cases) and define the inner product as before, taking into account that any element in the v.s. is

    a FINITE lin. comb. of some elements in the basis.

    Tonio
    Thank you very much! Great help. :-)
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