Infinite dimensional vectorial (dot) product)

[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.

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

 

[HallsofIvy]

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

 

[tacman]

The concept of a vectorial (dot) product in an infinite dimensional function space is not as straightforward as it is in a finite dimensional vector space. In a finite dimensional space, we can easily define the dot product as the sum of the products of the corresponding components of two vectors. However, in an infinite dimensional function space, this approach is not applicable as there are infinitely many components to consider.

Instead, the dot product in this context is defined using an integral. This integral, as mentioned in the question, takes the form of \int_{a}^{b}dx f^{*}(x)g(x), where f^{*}(x) is the complex conjugate of f(x) and g(x) is the other function in the product. This integral gives us a scalar value, which can be interpreted as the "projection" of one function onto the other.

But what about the vectorial product? In a finite dimensional space, the vectorial product is defined as a vector that is orthogonal to the two given vectors. However, in an infinite dimensional function space, the concept of orthogonality is not as clear. There are different ways to define orthogonality in this context, and it may not always be possible to find a vector that is orthogonal to two given functions.

In the question, the dual basis \tilde{\omega}^i is defined using the cross product of two vectors in a finite dimensional space. However, this definition may not be applicable in an infinite dimensional function space. Instead, we may need to define a different operation that satisfies the properties of a vectorial product in this context.

In conclusion, the concept of a vectorial product in an infinite dimensional function space is not as well-defined as it is in a finite dimensional vector space. The traditional definition of the cross product may not be applicable, and alternative approaches may need to be considered.