{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chemical Engineering Hand Written_Notes_Part_26

Chemical Engineering Hand Written_Notes_Part_26 - 54 2...

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 2

Chemical Engineering Hand Written_Notes_Part_26 - 54 2...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online