(6) We define the set of compactly supported sequences by qo = {(zn} : there exists some N > 0 so...
Im wondering how to do b). (6) We define the set of compactly supported sequences by qo = {(zn} : there exists some N > 0 so that Zn = 0 for all n >N). We define the set of compactly supported rational sequences by A={(za) E ao : zn E Q for all n E N). (a) Prove that A is countable (b) Prove that for 1 S p<oo the set A is dense in P. You may use...
the set of compactly supported sequences is defined by c00 = {{xn} : there exists some N ≥ 0 so that xn = 0 for all n ≥ N } Prove that for 1 ≤ p ≤ ∞ the metric space (c00, dp) is not complete.
(2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact. (b) Prove that for any є > 0 there exists some N > 0 so that for any x E A we have (c) Prove that A is totally bounded. (d) Prove that A is compact (2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact....
(2) Define the set A by (a) Prove that for any N 20 the set is compact. (b) Prove that for any e>0 there exists some N 2 0 so that for any x A we have (c) Prove that A is totally bounded. d) Prove that A is compact.
all parts A-E please. Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some number in (0, 1). Let f(n) :-0.aina2na3n , where ain 1s the first digit in the decimal form for the image of n, a2 is the second digit, and so on. If f (n) terminates after k digits, then our convention will be...
real analysis hint 9 Let co , a, and 〈æ be the Banach spaces consisting of all complex sequences x={ i-1, 2, 3,..., defined as follows: X E if and only if II x11 if and only if lxsup lloo. for which ξί (a) If y = {nJ E 11 and Ax = Σ ζίηǐ for every x ε co, then Λ is a bounded linear functional on (More precisely, these two spaces are not equal; the preceding statement exhibits...
6. Fix b (a) If m, n, p, q are integers, n > 0, q > 0, and r = mln-plg, prove that Hence it makes sense to define y (b")1/n. (b) Prove that b… = b,b" if r and s are rational. (c) If x is real, define B(x) to be the set of all numbers b', where t is rational and tSx. Prove that b-sup B(r) ris rational. Hence it b" = sup B(x) for every realx (d)...
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We define the degree of the constant polynomial p0 to be -. (A polynomial over R defines a function p : R R.) (a) Define a relation on the set of polynomials by p if and only if p(0) (0)...
2. (a) Let 11 = 0 and Zn+1=2r" +1 for all n E N. In +2 i. Find 2, , and ii. Prove that (r converges and find the value of its limit (b) Let a-V2, and define @n+1 = V2+@n for all n 1. Prove that lim an exists and equals 2 Hint: For both parts try to apply the Monotone Convergence Theorem
1. [4-+6+6-16 points Let /°0 denote the vector space of bounded sequences of real numbers, with addition and scalar multiplication defined componentwise. Define a norm Il on by Il xl = suplx! < oo where x = (x1,x2, 23, . .. ) iEN (a) Prove that is complete with respect to the norm | . (b) Consider the following subspaces of 1o i) c-the space of convergent sequences; (i) co-the space of sequences converging to 0; (iii) coo- the space...