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.
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.