Thesis (M.A.)--Boston University / This paper, utilizing the properties of Vector spaces, describes an approach to polynomial approximations of functions defined analytically or by a set of observations over some interval. If the function and its approximation are both considered tobe elements of a linear normed vector space, a weighted sum or integral of the square of the discrepancy between the function and its approximation is to be a minimum. When this condition is satisfied, and depending upon the interval of interest, the polynomial approximation to the function becomes either the Legendre, Chebyshev, Laguerre, or hermite approximation formulas. An investigation into the properties and applications of these formulas is included, and it is shown that these formulas give the best polynomial approximations to certain functions in the sense of least squares.