The final is located on page 3. You only have to print that page. Don't forget to read the instructions on the separate cover sheet! I created the file this way, so you did not have to avert your eyes
MIDTERM FOR MA108A DUE DATE: 4PM, TUESDAY NOVEMBER 2, 2010
Read the instructions on the separate sheet before you begin! (1) Let C 1 [0, 1] be the vectorspace of all continuously differentiable functi
MIDTERM FOR MA108A DUE DATE: 4PM, TUESDAY NOVEMBER 2, 2010
Read the instructions on the separate sheet before you begin! (1) Let C 1 [0, 1] be the vectorspace of all continuously differentiable functi
Math 108a 2011 Assignment 7 1. Let Bn = fx 2 X : jjAxjj n for all A 2 Fg. Since supA2F jjAxjj < 1 for all x 2 X, we have X = [1 Bn . Each Bn is closed as intersection of closed sets Bn = \A2F fx 2 n=1
MAT 108A-HW7 November 21. 2011. Due date: Tuesday November 29. 2011. at 4pm 1. (20)(Principle of uniform bound) Let X be a Banach space, and let Y be a normed vector space. Let F B(X; Y ). Show that i
Math 108a 2011 Assignment 6 1. (a) It is enough to con.rm claim for positive decreasing sequence of functions (fn )1 n=0 with fn (x) fn+1 (x) for all x 2 X which goes to 0. Let " > 0 and consider open
MAT 108A-HW6 November 15. 2011. Due date: Tuesday November 22. 2011. at 4pm 1. (20) (a) Let (X; d) be a compact metric space, and suppose that (fn )1 is a pointwise increasing sequence n=0 in C(X), i.
Math 108a 2011 Assignment 5 1. Without loss of generality we may assume that W1 W2 :.Let d be a metric induced by norm. We may take w1 2 W1 such that w1 6= 0. Then there is an w2 2 W2 nW1 1 . Then the
MAT 108A-HW5 November 8. 2011. Due date: Tuesday November 15. 2011. at 4pm 1. (20)Let V be an in.nite dimensional normed vector space, and suppose that V = Wn is a .nite dimensional vector subspace of
Math 108a 2011 Assignment 4 1. Without loss of generality we may assume that closed set F = XnG is nonempty. We de.ne metric on G with (x; y) = d(x; y) + j 1 d(x; F ) 1 j. d(y; F )
It is straightforwa
MAT 108A-HW4 November 1. 2011. Due date: Tuesday November 8. 2011. at 4pm 1. (20)Let (X; d) be a complete metric space and let G X be an open subset. Show that there is a complete metric on G such tha
MAT 108A-HW3 October 18. 2011. Due date: Tuesday October 25. 2011. at 4pm 1. (20)How many dierent sets can you get starting with a set A in some metric space and using operations c (complement), (clos
MAT 108A-HW2 October 11. 2011. Due date: Tuesday October 18. 2011. at 4pm 1. (20)Recall that every point x in the Cantor set De.ne function f : [0; 1] ! [0; 1] such that: (a) f (x) =
i=1 1 P bn 3n
on
MAT 108A-HW1 October 4. 2011. Due date: Tuesday October 11. 2011. at 4pm 1. (20)We say that poset (P; ): (a) satis.es ( ) i for all a < b in P , all maximal chains from a to b have the same length. (b
FINAL FOR MA108A DUE DATE: 4PM, FRIDAY DECEMBER 10, 2010
(1) Suppose f : X Y is a continuous map from a compact metric space X to an arbitrary metric space Y . (a) Show that for any A X we have f (cl(