7 ms·
My friend, you don't even need it to be in vector space for functional analysis. Truly what is needed is just an inner product. I will grant you the inner prod
by hayasaki 3y ago
My friend, you don't even need it to be in vector space for functional analysis. Truly what is needed is just an inner product. I will grant you the inner product must be linear and hence in a vector space.
- trostaft 3y agoWhy even require an inner product! You can get away with a lot just sitting in an Banach space (only a norm required).
- thorel 3y agoI agree. The GP comment contains some inaccuracies: most of the spaces of functions considered in functional analysis do not have an inner product defined on them, but are still vector spaces. The existence of an inner product presupposes a vector space structure, but the converse is not true… Perhaps the most famous example is provided by the Lp spaces [1] consisting of functions whose pth power is absolutely integrable. For p≥1, these spaces are Banach spaces (complete normed spaces) but it is only when p=2 that the norm is associated with an inner product. [1] https://en.wikipedia.org/wiki/Lp_space https://en.wikipedia.org/wiki/Lp_space
- eigenket 3y agoI dont understand the point of this comment. You obviously need it to be a vector space before you can define an inner product. Inner product spaces are very special examples of vector spaces.