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$V$ denotes real vector space $V$ consisting of all polynomials with real coefficients/any degree so $V$ is infinite dimensional. A symmetric bilinear form is defined on V by

$$(f,g) = \int_0^\infty f(x)g(x)e^{-x} dx$$

i) State briefly why this is positive definite on $V$ (may quote any general property of real integrals).

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What part of the definition of "positive definite" is causing trouble? – Nate Eldredge Apr 24 '12 at 18:52
Hint: a definite integral of a nonnegative function is non-negative. – rschwieb Apr 24 '12 at 19:29

A bilinear form $b$ is positive definite if $b(v,v) > 0$ if $v \neq 0$.

So you want to show that $(p,p) = \int_0^\infty p^2(x) e^{-x} dx > 0$ for $p(x) \neq 0$.

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