assignment5

assignment5 - X can be written as the intersection of...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 255 ASSIGNMENT 5 This assignment is due in class on Monday, February 14 Problems Please justify carefully your answers. 1. [10 points] Let ( X,d X ) and ( Y,d Y ) be metric spaces and f : X Y . Prove that the following statements are equivalent. (1) f is continuous on X . (2) For any open set V Y , the set f - 1 ( V ) = { x X : f ( x ) V } , is open in X . 2. [25 points] Let X be a metric space and let E be a nonempty subset of X . Define d E ( x ) = inf { d ( x,e ) : e E } . (1) Show that d E ( x ) d ( x,y ) + d E ( y ). (2) Show that | d E ( x ) - d E ( y ) | ≤ d ( x,y ) and deduce that d E : X R is a continuous function. (3) If in addition E is a closed subset of X , show that d E ( x ) = 0 x E . (4) Deduce from (2) that for fixed t > 0, { x : d E ( x ) < t } is an open subset of X containing E . (5) Choosing t = 1 /n , n N in (4) show that every closed subset of
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: X can be written as the intersection of countably many open subsets of X . 3. [10 points] Let ( X,d ) be a complete metric space and f : X X a continuous function such that for some integer N &gt; 0, f N = f f f | {z } N times , is a contraction on X . Prove that there exists unique x X such that f ( x ) = x . 4. [10 points] Consider Banach space C ([0 , 1]). Let F : C ([0 , 1]) C ([0 , 1]) be a function dened by F ( f ) = sin( f ) . In other words, ( F ( f ))( t ) = sin( f ( t )) for t [0 , 1]. Prove that F is a continuous function. 1...
View Full Document

Ask a homework question - tutors are online