## Defining A Vector Space (resp. Inner Product Space) Structure On a Set of Functions Whose Images Lie In a Vector Space (resp. Inner Product Space)

**Point of post: **Just as the title suggests in this post we shall describe the natural way to imbue the set , where is some vector space and is a set, with a vector space structure (the usual one). Moreover, we define a natural way to define an inner product on when happens to itself be an inner product space.

*Motivation*

In this post we discuss something which by itself is not very profound. Indeed, we discuss the natural way to turn the set (alternatively denoted ) where is a set and is a -space with a -space structure. No surprises here, we will just define the addition and scalar multiplication pointwise. As I said, there is nothing deep here. This will just be a convenient post for when I say “Give the ‘usual’ structure’ I can make precise what I mean by this. Slightly less banal we shall discuss a natural (one among many) ways to define an inner product structure on assuming has an inner product structure and is finite.

*Giving a Vector Space Structure*

Suppose that is a set and a -space. Let, as usual, (alternatively ) denote the set of all functions . There is a natural way to define a -space structure on . Indeed, we can define the *sum *of to be the function defined by for every . Similarly, we can define the *product *of and to be the function defined by . It is evident that this defines a -space structure on .

**Defining an Inner Product Structure On **

A little less obvious is if is an inner product space over (where is equal to either or , but more general definitions will hold) and is a finite set. In particular, let denote the inner product on . Then, if is given the usual vector space structure as defined above we can augment it with an inner product given by

It’s obvious that this is an inner product. Often this inner product is accompanied by some normalization constant. Often, when it is defined by

When and is reduced to the usual inner product this reduces to

This was the case when we defined the inner product on the group algebra.

We denote that the above construction actually generalizes. Namely, if is any set and we define to be the set of all with finite support then is evidently an inner product space with the above inner product.

**References:**

1. Roman, Steven. *Advanced Linear Algebra*. New York: Springer-Verlag, 1992. Print.

[…] be a finite group and . Let then be a representation of . Consider then the set with the usual inner product structure with normalization constant . Define then to be the set . We claim then that . Indeed, let and […]

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