Im wondering how to do b). (6) We define the set of compactly supported sequences by...
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.
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...
problem 23 please :) and here is Q.21 Problem 23. Recall from Problem 21 the equivalence relation ~ on the set of rational Cauchy sequences C. Define 〈z) E C to be eventually positive if there is an M є N such that xn > 0 for all Prove that eventually positive is a well defined notion on c/ (z〉 ~ 〈y), then 〈y〉 İs eventually positive. ie. if 〈z) is eventually positive and Problem 21. Let C be the...
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)...
(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.
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...
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)...
Galois Theory chapter 2 DUP . 2.2 A formal definition of C[t runs as follows. Consider the set of all infinite sequences (an)neN = (20, 21,..., Ans...) where an EC for all n E N, and such that an = 0 for all but a finite set of n. Define operations of addition and multiplication on S by the rules (an) + (bn) = (un) where Un = an + bn (an) (sn) = (vn) where yn = anbo +...