Abstract Nonsense

Crushing one theorem at a time

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.

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April 21, 2011 - Posted by | Algebra, Linear Algebra | , ,

1 Comment »

  1. […] 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 […]

    Pingback by Induced Representations (Pt. I) « Abstract Nonsense | April 23, 2011 | Reply


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