# Hilbert space on a finite set

If X is a finite set, what does the Hilbert space $L^2(X)$ means? - saw this notion on The Princeton Companion to Mathematics.

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Which bit of the definition of Hilbert space is causing confusion here? –  user16299 Sep 27 '11 at 8:13

If $X$ contains, say, $n$ elements then $L^2(X)$ reduces to the ordinary Euclidean space $\mathbb{R}^n$. EDIT: As commenter points out, here we have implicitly assumed that $X$ is equipped with the counting measure (call it $m$), that is, every subset of $X$ has a measure equal to its cardinality. This way the integral
$$\int_X\lvert f(x) \rvert^2\, dm$$
$$\sum_{k=1}^n \lvert f(k)\rvert^2.$$
I think what's missing is the explicit mention of a measure. So you should say that you take counting measure, for example. Just give some positive weight to each point in $X$, otherwise your $L^2$-space won't have dimension $n$. –  commenter Sep 27 '11 at 8:19