Benefits of function space What are the benefits of having the concept of function space? I'm not sure if I understand the concept itself but is it that you can write a graph of some set of functions? If so, is one of the benefits to have functional space that you can find something from the angle of a curve that that graph makes? Or does it exist just for generalizing any set of functions like the power set of a set X or all possible functions that map a set X to another set Y? This must be a stupid question....Sorry!
 A: Remember when you started dealing with numbers and you were presented the concept of function. You could've asked yourself "What are the benefits of considering functions that take numbers into numbers?" They turned out to be very useful right?
Well, with function spaces the situation is the same:
We want to understand better functions that take numbers into numbers (for a start), so we hope that the same strategy works here; we make these functions that take numbers into numbers play the role of numbers and consider functions that take functions into numbers (functionals). I.e., we know how useful it is the study of functions of numbers towards a deeper understanding of numbers, so we expect the study of functions of functions to provide insight toward a deeper understanding of  functions.
So, what are the benefits of having the concept of function spaces? That our hopes are fully rewarded!!
Of course, since functions are more complicated objects than numbers, a lot of difficulties arise when trying to make them play that role, the lost of a great deal of visualization, intuition and familiarity being the most obvious ones. But hei, what fun would it be if it were so easy?!
