Chemical Engineering Hand Written_Notes_Part_17

Chemical Engineering Hand Written_Notes_Part_17 - 36 2...

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36 2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS h x + y , x + y i = h x , x i + h x , y i + h y , x i + h y , y i . (4.10) h x , x i +2 |h x , y i| + h y , y i (4.11) h x , x i +2 p h x , x ih y , y i + h y , y i (4.12) (4.13) p h x + y , x + y i p h x , x i + p h y , y i Thus, the candidate function p h x , x i satis f es all the properties necessary to de f ne a norm, i.e. p h x , x i 0 x X and p h x , x i =0 iff x = 0 (4.14) p h α x x i = | α | p h x , x i (4.15) p h x + y , x + y i p h x , x i + p h y , y i (Triangle inequality) (4.16) Thus, the function p h x , x i indeed de f nes a norm on the inner product space X. In fact the inner product de f nes the well known 2-norm on X, i.e. (4.17) k x k 2 = p h x , x i and the triangle inequality can be stated as (4.18) k x + y k 2 2 k x k 2 2 +2 k x k 2 . k y k 2 + k y k 2 2 . =[ k x k 2 + k y k 2 ] 2 (4.19) or k x + y k 2 k x k 2 + k y k 2 Definition 14 . ( Angle ) The angle θ between any two vectors in an inner
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Chemical Engineering Hand Written_Notes_Part_17 - 36 2...

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