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.
I 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.
I 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.