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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!

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    As is, the question is not very well-posed... if you're familiar with point-set topology, then you know what the utility of having a notion of a topological space is, and what kinds of properties it lets you describe. The collection of all functions can also be topologized, and so for the same reason, we may want to describe these properties. – 2017-02-04
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    Search for "history of functional analysis" and you may find readings on how and why people started studying function spaces. Along with what Alfred Yerger says, if you're solving problems that have unknown functions in equations, like integral or differential equations, building tools for solving these equations out of a specified set of functions is easier if you have extra structure, such as the tools of linear algebra, metric spaces, etc. – 2017-02-04
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    @AlfredYerger I feel I took the term "space" wrong....I took it as a general meaning "space". I thought it meant to be a space where you can graph something on like the Cartesian coordinate plane. So it basically refers to a set when used as a math term and doesn't have anything to do with graphs particularly? – 2017-02-04
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    @stacko: Yes, it refers to a set with some additional structure. E.g., vector space, topological space, metric space, normed space. In my experience function spaces are typically topological vector spaces, but "space" is such a broad term that one could plausibly study "function spaces" consisting of any set of functions with whatever additional structure that may be relevant. – 2017-02-04
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    A space in math usually means something more general than just the plane or something, although this is one example. A space has to have properties that tell you what kind of parts it has, and how they are put together. This is a thoroughly explored idea in the realm of 'topology.' – 2017-02-04

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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?!