docsamsol

# docsamsol - Analysis PhD Examination Sample Solutions 1....

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

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.

Unformatted text preview: Analysis PhD Examination Sample Solutions 1. Let X and Y be normed linear spaces. Prove that the normed space B ( X, Y ) (comprising all bounded linear maps from X to Y ) is complete if and only if Y is complete. Solution ( ⇐ ) This is the ‘familiar’ direction. Let ( T n : n > 0) be Cauchy in B ( X, Y ). If x ∈ X then k T p ( x )- T q ( x ) k 6 k T p- T q kk x k so that ( T n ( x ) : n > 0) is Cauchy in Y ; as Y is complete, we may define T ( x ) := lim n T n ( x ). Plainly, T : X → Y is linear. Let ε > 0 and choose N > 0 so that if p, q > N then k T p- T q k 6 ε : if k x k 6 1 then k T p ( x )- T q ( x ) k 6 ε and passage to the limit as q → ∞ yields k T p ( x )- T ( x ) k 6 ε ; thus k T p- T k 6 ε when p > N and so T p → T in B ( X, Y ). ( ⇒ ) Let ( y n : n > 0) be Cauchy in Y ; of course, we must pass to B ( X, Y ) by some means. One such: fix (0 6 =) a ∈ X and choose (how?) φ ∈ X * such that φ ( a ) = 1. For n > 0 define T n ∈ B ( X, Y ) by x ∈ X ⇒ T n ( x ) = φ ( x ) y n . Note that k T p ( x )- T q ( x ) k = | φ ( x ) |k y p- y q k 6 k y p- y q kk φ kk x k so ( T n : n > 0) is Cauchy. Let T = lim n T n...
View Full Document

## This note was uploaded on 11/09/2011 for the course MAA 5228 taught by Professor Robinson during the Fall '09 term at University of Florida.

### Page1 / 3

docsamsol - Analysis PhD Examination Sample Solutions 1....

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

View Full Document
Ask a homework question - tutors are online