> (well, certain classes of functions, such as L^2 anyway -- I'm certainly no expert here) For any vector space V over a field F, the set of functions V -> V i

> (well, certain classes of functions, such as L^2 anyway -- I'm certainly no expert here) For any vector space V over a field F, the set of functions V -> V is a vector space over F. This is essentially because the vector space properties are carried over "point-wise". When you consider only certain classes of functions (such as L^2), the question of whether they form a vector space boils down to the question whether those classes are closed under addition and scalar multiplication. All other vector space properties are satisfied because they are in the larger vector space of all functions.