Chemical Engineering Hand Written_Notes_Part_19

Chemical Engineering Hand Written_Notes_Part_19 - 40 2....

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2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS where W is a positive de f nite matrix. Then, length of ° ° x (1) ° ° W, 2 = 6 and the unit vector b e (1) becomes (4.40) b e (1) = x (1) k x (1) k W, 2 . = 1 6 0 1 6 The remaining two orthonormal vectors have to be computed using the inner product de f ned by equation 4.39. Example 31 . Gram-Schmidt Procedure in C[a,b] : Let X represent set of continuous functions on interval 1 t 1 with inner product de f ned as (4.41) h x ( t ) , y ( t ) i = 1 Z 1 x ( t ) y ( t ) dt Given a set of four linearly independent vectors (4.42) x (1) ( t )=1; x (2) ( t )= t ; x (3) ( t )= t 2 ; x (4) ( t )= t 3 we intend to generate an orthonormal set. Applying Gram-Schmidt procedure (4.43) e (1) ( t )= x (1) ( t ) k x (1) ( t ) k = 1 2 (4.44) ­ e (1) ( t ) , x (2) ( t ) ® = 1 Z 1 t 2 dt =0 (4.45) z (2) ( t )= t ­ x (2) , e (1) ® . e (1) = t = x (2) ( t ) (4.46) e (2) = z (2) k z (2) k (4.47)
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Chemical Engineering Hand Written_Notes_Part_19 - 40 2....

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