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, while printing the final ;-).
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FINAL FOR MA108A DUE
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 functions on the interval [0, 1]. We know that f = supx[0,1]
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 functions on the interval [0, 1]. We know that f = supx[0,1]
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 X : jjAxjj ng. Since every Banach space is of the seco
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 if for every x 2 X we have supA2F jjAxjj < 1 then supA2F
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 sets Un = fx 2 X : fn (x) < "g. Collection fUn : n 0g
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.e. fn (x) fn+1 (x) for all x 2 X and all n 0. If pointw
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 there is an w3 2 W3 nW2 such that such that d(w1 ; w2 ) <
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 V . Prove that V is not complete.
1 S
Wn , where each
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 straightforward to check that this is a metric. Since d(x; y) (x; y)
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 that it has the same open sets as G has with relative metr
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), (closure), (interior)? 2. (20)Let c be the set of convergent
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 [0; 1] can be written as
i=1
1 P
2bn 3n
where bn 2 f0;
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) is graded by function g : P ! Z ( from P into the set
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(A) = cl(f (A). Proof. As X is compact, and cl(A) is a c