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Infinite dimensional vectorial (dot) product)

  1. Jul 14, 2007 #1
    If for a function space , we can define the scalar product of any given 2 functions as

    [tex] \int_{a}^{b}dx f^{*}(x)g(x) [/tex]

    what happens with its analogue the 'dot/cross' (vectorial product of 2 vectors which is itself another vector orthogonal to the 2 given ones)

    the question is not in vane since to define the dual basis of any vector

    [tex] \tilde{\omega}^i = {1 \over 2} \, \left\langle { \epsilon^{ijk} \, (\vec e_j \times \vec e_k) \over \vec e_1 \cdot \vec e_2 \times \vec e_3} , \qquad \right\rangle [/tex]

    hence for a function space we would find that.

    [tex] h(x) = (??) f(x) \times g(x) [/tex]
  2. jcsd
  3. Jul 14, 2007 #2


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    Science Advisor

    The cross product is only defined in 3 dimensions so your question does not appear to me to be "well defined".
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