Chemical Engineering Hand Written_Notes_Part_15

Chemical Engineering Hand Written_Notes_Part_15 - 32 2...

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32 2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS When we are working in R n or C n , all convergent sequences are Cauchy sequences and vice versa. However, all Cauchy sequences in a general vector space need not be convergent. Cauchy sequences in some vector spaces exhibit such strange behavior and this motivates the concept of completeness of a vector space. Definition 12 . (Banach Space): A normed linear space X is said to be complete if every Cauchy sequence has a limit in X . A complete normed linear space is called Banach space. Examples of Banach spaces are ( R n , k . k 1 ) , ( R n , k . k 2 ) , ( R n , k . k ) ( C n , k . k 1 ) , ( C n , k . k 2 ) , ( l , k . k 1 ) , ( l , k . k 2 ) etc. Concept of Banach spaces can be better understood if we consider an example of a vector space where a Cauchy sequence is not convergent, i.e. the space under consideration is an incomplete normed linear space. Note that, even if we f nd one Cauchy sequence in this space which does not converge, it is su cient to prove that the space is not complete.
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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_15 - 32 2...

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