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 f 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 f nite sequences ( l ) with p-norm: An element in this space,
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical Engineering Hand Written_Notes_Part_14 - 30 2....

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