midterm1-practice

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

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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 A9 ¥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 ¡ A9 ¥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 ` A9 ¥BVU¨ 7 6 h# D ¦ igfR# c d ` ¦&# X X # Y! • ˜# …6‰ 1—–… % ` 0 ƒ „¡ 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 A9 ¢B@¨ ! ¡ • % #! ”b2&Y”¡ … ¥¥@U¨ A9 A9 ¥¥@U¨ c ˆ# c `  # A ...
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This note was uploaded on 09/19/2011 for the course MATH 131B taught by Professor Hitrik during the Winter '08 term at UCLA.

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