- #1
Klaus_Hoffmann
- 85
- 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]
[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]