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Vector Space A vector space V is a set that is closed under finite vector addition and scalar multiplication. The basic example is ndimensional Euclidean space , where every element is represented by a list of n real numbers, scalars are real numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately. Basis A basis of a vector space V is defined as a subset span V. Consequently, if can be uniquely written as of vectors in V that are linearly independent and vector space is a list of vectors in V, then these vectors form a basis if and only if every where , ..., are elements of or . A vector space V will have many different bases, but there are always the same number of basis vectors in each of them. The number of basis vectors in V is called the dimension of V. Every spanning list in a vector space can be reduced to a basis of the vector space. The span of subspace generated by vectors and is Complete Basis A set of orthogonal functions is termed complete in the closed interval function f(x) in the interval, the minimum square error if, for every piecewise continuous (where denotes the L2-norm with respect to a weighting function w(x)) converges to zero as n becomes infinite. Symbolically, a set of functions is complete if where the above integral is a Lebesgue integral. Examples of complete orthogonal systems include special type of system known as a complete biorthogonal system), over (which actually form a slightly more Orthogonal functions Two functions f(x) and g(x) are orthogonal over the interval with weighting function w(x) if (1) If, in addition, (2) (3) the functions f(x) and g(x) are said to be orthonormal. Coefficients Error