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Chemical Engineering Hand Written_Notes_Part_14

# Chemical Engineering Hand Written_Notes_Part_14 - 30 2...

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30 2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS (1) k x k 0 for all x X ; k x k = 0 if and only if x = 0 ( zero vector ) (2) k x + y k k x k + k y k for each x , y X. ( triangle inequality ). (3) k α x k = | α | . k x k for all scalars α and each x X The above de fi nition of norm is an abstraction of usual concept of length of a vector in three dimensional vector space. Example 23 . Vector norms: (1) ( R n , k . k 1 ) : Euclidean space R n with 1-norm: k x k 1 = N P i =1 | x i | (2) ( R n , k . k 2 ) : Euclidean space R n with 2-norm: k x k 2 = " N X i =1 ( x i ) 2 # 1 2 (3) ( R n , k . k ) : Euclidean space R n with ∞− norm: k x k = max | x i | (4) ³ R n , k . k p ´ : Euclidean space R n with p-norm, (3.1) k x k p = " N X i =1 | x i | p # 1 p , where p is a positive integer (5) n -dimensional complex space ( C n ) with p-norm, (3.2) k x k p = " N X i =1 | x i | p # 1 p , where p is a positive integer (6) Space of in fi nite sequences ( l ) with p-norm: An element in this space, say x l , is an in fi nite sequence of numbers (3.3) x = { x 1 , x 2 , ........ , x n , ........ } such that p-norm is bounded (3.4) k x k p = " X i =1 | x i | p # 1 p <

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Chemical Engineering Hand Written_Notes_Part_14 - 30 2...

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