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**HallsofIvy** Then you probably learned that there are many more vector spaces than just $\displaystyle R^n$.

For example, the set of all polynomials, p(x), of degree n or less (for fixed n), is a vector space. We can define an "inner product" to be $\displaystyle <p, q>= \int_0^1 p(x)q(x)dx$.

More generally, the vector space, $\displaystyle L_2[a, b]$, of functions, f(x), defined on [a, b] such that the (Lebesque) intgral $\displaystyle \int_a^b f^2(x)dx$ exists, is an infinite dimensional vector space. And we can define the inner product as [itex]<f, g>= \int_a^b f(x)g(x)dx[/tex]. The "length" of such a vector would be $\displaystyle \sqrt{\int_a^b f^2(x)dx}$

$\displaystyle L_1[a,b]$, the set of all functions, f(x), defined on [a, b], such that $\displaystyle \int_a^b |f(x)|dx$ exists also forms a vector space but one on which we **cannot** define an inner product on this space. We can define the length of a vector to be $\displaystyle \int_a^b|f(x)|dx$.