Chemical Engineering Hand Written_Notes_Part_26

Chemical Engineering Hand Written_Notes_Part_26 - 54 2....

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54 2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS (18) Show that in C[a,b] with maximum norm, we cannot de f ne an inner product h x , y i such that h x , x i 1 / 2 = k x k . In other words, show that in C [ a, b ] the following function h f ( t ) ,g ( t ) i = max t | x ( t ) y ( t ) | cannot de f ne an inner product. (19) In C (1) [ a, b ] is h x , y i = b Z a x 0 ( t ) y 0 ( t ) dt + x ( a ) y ( a ) an inner product? (20) Show that in C (1) [ a, b ] is h x , y i = b Z a w ( t ) x ( t ) y ( t ) dt with w ( t ) > 0 de f nes an inner product. (21) Show that parallelogram law holds in any inner product space. k x + y k 2 + k x y k 2 =2 k x k 2 +2 k y k 2 Does it hold in C[a,b] with maximum norm? (22) The triangle inequality asserts that, for any two vectors x and y belonging to an inner product space k x + y k 2 || y || 2 + || x || 2 After squaring both the sides and expanding , reduce this to Schwarz in- equality. Under what condition Schwarz inequality becomes an equal- ity? (23) Show that operator d/dx ( . ): C (1) [ a, b ] C [ a, b ] is onto C [ a, b ] but not one to one.
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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_26 - 54 2....

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