Weierstrass's Theorem states that any function continuous over a closed interval can be written in terms of a Cauchy sequence of polynomials that converge uniformly to that function. I will not give a proof since the one in front of me is tedious.

Weierstrass's Theorem differs from Taylor's Theorem in a few important ways. While Taylor's Theorem requires analyticity (infinite differentiability) of a function, Weierstrass's Theorem requires only continuity over an interval. However, if a function satisfies the analyticity requirement of Taylor's Theorem, then it can be written as a power series--this means that one gets better and better polynomial approximations by adding higher-order terms, while leaving the lower-order coefficients fixed. Weierstrass's Theorem, while more general, does not guarantee a uniformly convergent power series.

Weierstrass's Theorem implies that any (e.g. discontinuous) function in Hilbert Space can be written in terms of a sequence of polynomials that converge in the mean to that function. Thus Weierstrass's Theorem paves the way for the construction of sets of orthonormal functions that are complete over Hilbert Space (e.g. Legendre Polynomials).

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