appendixa_sol

# appendixa_sol - Exercises from Appendix A A.12 Hints Let s...

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: Exercises from Appendix A A.12 Hints: . Let s N = N n =1 f n , and show that the sequence of partial sums { s N } N N is Cauchy in X. . Suppose that every absolutely convergent series is convergent. Let { f n } n N be a Cauchy sequence in X. Show that there exists a subsequence { f n k } k N such that bardbl f n k +1 f n k bardbl &amp;lt; 2 k for every k (see Problem A.2). Then k ( f n k +1 f n k ) is absolutely convergent, hence converges, say to f. Show that { f n } n N has a subsequence that converges (consider the partial sums of k ( f n k +1 f n k )). Show { f n } n N converges (consider Problem A.1). A.17 Hints: There are several ways to prove the Triangle Inequality for 1 &amp;lt; p &amp;lt; . One way is to begin with bardbl x + y bardbl p p = summationdisplay k I | x k + y k | p = summationdisplay k I | x k + y k | p 1 | x k + y k | summationdisplay k I | x k + y k | p 1 | x k | + summationdisplay k I | x k + y k | p 1 | y k | . Then apply H olders Inequality to each sum using the exponent p on the first factor and p for the second (recall that p = p/ ( p 1)). Then divide both sides by bardbl x + y bardbl p 1 p . To show completeness, consider Problem A.6. Show that if { x n } n N is a Cauchy sequence in p and we write x n = ( x n (1) ,x n (2) ,... ) , then for each fixed k we have that { x n ( k ) } n N is a Cauchy sequence of scalars, hence con- verges. This gives a candidate sequence x for the limit of { x n } n N . Use the fact that { x n } n N is Cauchy together with the componentwise convergence to show that bardbl x x n bardbl p . A.18 Hint: Assume 0 &amp;lt; p &amp;lt; 1 . Show that (1 + t ) p 1 + t p for t &amp;gt; , and use this to show that ( a + b ) p a p + b p for a , b . A.20 Hints: To show that C c ( R ) is not complete, choose a function in C ( R ) that is nonzero everywhere, and then define appropriate continuous functions g N supported on [ N 1 ,N + 1] such that g N = g on [ N,N ] . Show that { g N } N N is a Cauchy sequence in C c ( R ) that does not converge within C c ( R ) with respect to the uniform norm. On the other hand, this sequence does converge in C ( R ) ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

appendixa_sol - Exercises from Appendix A A.12 Hints Let s...

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

View Full Document
Ask a homework question - tutors are online