midterm1-practice

# midterm1-practice - MATH 131B 1ST PRACTICE MIDTERM Problem...

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

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

Unformatted text preview: MATH 131B 1ST PRACTICE MIDTERM Problem 1. State the book’s deﬁnition of: (a) Uniform convergence of a sequence of function (b) A metric (c) Pointwise convergence of a sequence of functions (d) A continuously differentiable function of two variables (e) A continuous function of two variables (f) The supremum norm ¤ ¡ ¥£¢ Problem 2. Prove that a uniform limit of a sequence of continuous functions is also continuous.  © ¨ ¦ §¡ be a sequence of functions that converges uniformly on ¡ to a function .     4 # '% #! 5)3&\$2¡ Problem 3. Let Show that 0 # '% #! ¦ 1)(&\$"§¡ Give an example which shows that the conclusion does not hold if we only assume pointwise convergence. be the space of functions on A9 ¥B@¨ Problem 4. Let Let which have a continuous derivative.  A  9¨ 7 ¥B@¢86 4 ¤ Q ¡ H ¤ ¡ D ¡ TGSR¢PIGC¢FEC¢ is a norm is complete in this norm. aD b# D C% if and if . is Cauchy for this metric if and only whenever . (c) Show that is 9 `  X ` A W e sr , but does not converge %# &Y! Q ¡ A9 ¥B@¨ 9 X e v B¦ w wv xB¦ D¦ ut¡ y 7 be a continuously differentiable function. Suppose that is a “uniform contraction”: there is a constant .   q 2p on W W converges pointwise to A9 6   A ¡9 ¢B@¨  5` 1 ` A9 ¥BVU¨ 7 6 h# D ¦ igfR# c d ` ¦&# X X # Y!  # 6 1 % ` 0  ¡ Problem 7. Let for all . Show that for all , # Y! Problem 6. Show that uniformly. D £% Problem 5. Endow with the following metric : (a) Show that is a metric. (b) Show that a sequence if it is eventually constant; i.e., for some , complete with respect to this metric. D 1# ¡ C¢ (a) Show that (b) Show that so that A9 ¢B@¨ ! ¡  % #! b2&Y¡  ¥¥@U¨ A9 A9 ¥¥@U¨ c # c `  # A ...
View Full Document

## This note was uploaded on 09/19/2011 for the course MATH 131B taught by Professor Hitrik during the Winter '08 term at UCLA.

Ask a homework question - tutors are online