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## Main Question or Discussion Point

I'm on a course which is currently introducing me to the concept of Hilbert spaces and the professor in charge was giving examples of such spaces. He ended by considering [tex]V[/tex], the space of polynomials with complex coefficients from [tex]\mathbb{R}[/tex] to [tex]\mathbb{C}[/tex]. He then, for [tex]f,g\in V[/tex], defined

and claimed - without proof - that [tex]V[/tex] equipped with [tex](\cdot ,\cdot )[/tex] is an inner product space, but that [tex]V[/tex] is

[tex](f,g)=\int_{0}^{\infty}f(x)\bar{g(x)}e^{-x}dx[/tex]

and claimed - without proof - that [tex]V[/tex] equipped with [tex](\cdot ,\cdot )[/tex] is an inner product space, but that [tex]V[/tex] is

__not__complete. Could anyone come up with a clever way of showing that this is true?