# Abstract Nonsense

## 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 $\mathscr{V}^X$, where $\mathscr{V}$ is some vector space and $X$ is a set, with a vector space structure (the usual one). Moreover, we define a natural way to define an inner product on $\mathscr{V}^X$ when $\mathscr{V}$ happens to itself be an inner product space.

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Motivation

In this post we discuss something which by itself is not very profound. Indeed, we discuss the natural way to turn the set $\mathscr{V}^X$ (alternatively denoted $\text{Map}(X,\mathscr{V})$) where $X$ is a set and $\mathscr{V}$ is a $F$-space with a $F$-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 $\mathscr{V}^X$ 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 $\mathscr{V}^X$ assuming $\mathscr{V}$ has an inner product structure and $X$ is finite.

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Giving $\mathscr{V}^X$ a Vector Space Structure

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Suppose that $X$ is a set and $\mathscr{V}$ a $F$-space. Let, as usual, $\mathscr{V}^X$ (alternatively $\text{Map}(X,\mathscr{V})$) denote the set of all functions $f:X\to\mathscr{V}$. There is a natural way to define a $F$-space structure on $\mathscr{V}^X$. Indeed, we can define the sum of $f,g\in\mathscr{V}^X$ to be the function $f+g$ defined by $(f+g)(x)=f(x)+g(x)$ for every $x\in X$. Similarly, we can define the product of $\alpha\in F$ and $f\in\mathscr{V}^X$ to be the function $\alpha f$ defined by $(\alpha f)(x)=\alpha f(x)$. It is evident that this defines a $F$-space structure on $\mathscr{V}^X$.

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Defining an Inner Product Structure On $\mathscr{V}^X$

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A little less obvious is if $\mathscr{V}^X$ is an inner product space over $\mathbb{F}$ (where $\mathbb{F}$ is equal to either $\mathbb{R}$ or $\mathbb{C}$, but more general definitions will hold) and $X$ is a finite set. In particular, let $\langle\cdot,\cdot\rangle_\mathscr{V}$ denote the inner product on $\mathscr{V}$. Then, if $\mathscr{V}^X$ is given the usual vector space structure as defined above we can augment it with an inner product $\langle\cdot,\cdot\rangle$ given by

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$\displaystyle \left\langle f,g\right\rangle=\sum_{x\in X}\left\langle f(x),g(x)\right\rangle_\mathscr{V}$

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It’s obvious that this is an inner product. Often this inner product is accompanied by some normalization constant. Often, when $\#(X)<\infty$ it is defined by

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$\displaystyle \left\langle f,g\right\rangle=\frac{1}{\#(X)}\sum_{x\in X}\left\langle f(x),g(x)\right\rangle_\mathscr{V}$

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When $\mathscr{V}=\mathbb{C}$ and $\langle\cdot,\cdot\rangle_\mathbb{C}$ is reduced to the usual inner product this reduces to

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$\displaystyle \left\langle f,g\right\rangle=\sum_{x\in X}f(x)\overline{g(x)}$

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

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We denote that the above construction actually generalizes. Namely, if $X$ is any set and we define $\mathscr{V}^X_0$ to be the set of all $f\in\mathscr{V}^X$ with finite support then $\mathscr{V}^X_0$ is evidently an inner product space with the above inner product. $\blacksquare$

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References:

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