David G. Luenberger. Optimizatio by Vector Space Methods. John Wiley &: Sons, New York, 1969.  Home/Search   Document Not in Database   Summary   Related Articles

....(x I y) o. Similarly, a vector x is said to be orthogonal to a set S C X, denoted x 2 S, if x 2 y for all y ff S. One of the most important properties of Hilbert spaces is that its elements can be approximated by projecting them onto some convenient subspace, typically one of finite dimension [25]. 3 Linear Operators A mapping T from a vector space X to a vector space Y is said to be linear if it satisfies T(x y) T(x) T(y) 6) for any x ff X, y Y, and any scalars ( and . In analogy with matrix notation, it is traditional to write Tx instead of T(x) if T is linear. Furthermore, ....

....can write this as for = 1, n. But, by equation (51) we have (55) 56) 58) Equation (58) consists of the familiar normal equations of the least squares method. The matrix of inner products is a Gram matrix , which is nonsingular provided that the functions , are linearly independent [25]. 5.4.3 The Galerkin Method The Galerkin method also requires the structure of a Hilbert space; in fact, its form is almost identical to the least squares method. The difference, again, is in the criterion for picking the approximation f. In the least squares method, we required the residual f ....

David G. Luenberger. Optimizatio by Vector Space Methods. John Wiley &: Sons, New York, 1969.

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