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It satisfies all the usual properties that the inner product for finite dimensional vectors in n does also. These properties are so common we will codify them into a definition for what an inner product for a vector space V should behave like. 1 (Real Inner Product) Let V be a vector space with the reals as the scalar field. Then a mapping ω which assigns a pair of objects to a real number is called an inner product on V if 1. ω(u, v) = ω(v, u); that is, the order is not important for any two objects.

Take any finite subset from W . Label the resulting powers as {n1 , n2 , . . , np }. Write down the linear dependence equation c1 t n1 + c2 t n2 + · · · + cp t np = 0. Take np derivatives to find cp = 0 and then backtrack to find the other constants are zero also. Hence C[0, 1] is an infinite dimensional vector space. It is also clear that W does not span C[0, 1] as if this was true, every continuous function on [0, 1] would be a polynomial of some finite degree. This is not true as sin(t), e−2t and many others are not finite degree polynomials.

I=1 This is a quadratic expression and setting the gradient of E to zero, we find the critical points aj =< u, w j > . This is a global minimum for the function E. Hence, the optimal p∗ has the form N ∗ p = < u, w i > wi . i=1 32 2 Graham–Schmidt Orthogonalization Finally, we see N ∗ < p − p , wj > = < p, wj > − < p, wk >< wk , wj > k=1 = < p, wj > − < p, wj >= 0, and hence, p = p∗ is orthogonal of W . 5 Graham–Schmidt Orthogonalization Let’s assume we are given two linearly independent vectors in MatLab/Octave and enter two such vectors.