Infinite dimensional analysis is essentially functional analysis. A crude analogy is that a function is an infinite dimensional vector [...f(0),f(0+epsilon),...

Infinite dimensional analysis is essentially functional analysis. A crude analogy is that a function is an infinite dimensional vector [...f(0),f(0+epsilon),...] so on. Although extremely interesting in its own right, this approach however will not give you a handle on how the behavior changes as the dimensionality grows.