# Infinite dimensional vectorial (dot) product)

1. Jul 14, 2007

### Klaus_Hoffmann

If for a function space , we can define the scalar product of any given 2 functions as

$$\int_{a}^{b}dx f^{*}(x)g(x)$$

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

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

hence for a function space we would find that.

$$h(x) = (??) f(x) \times g(x)$$

2. Jul 14, 2007

### HallsofIvy

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