Homework Answers
in last part take any
neighbiurhood of any element of A with radius 0.5 it contains only
one point of A so it is not convergent sequnce
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Topology
b) Let S denote the subset of co consisting of sequences with rational entries of...
Topology (b) Let S denote the subset of co consisting of sequences with rational entries of which at most finitely many are nonzero. (i) Show that S is dense in co with the sup norm. [Hint: Show that...
Topology
(b) Let S denote the subset of co consisting of sequences with rational entries of which at most finitely many are nonzero. (i) Show that S is dense in co with the sup norm. [Hint: Show that for every r E co and every ε > 0, there exists y S such that llx-yI100 < ε.j (ii) Conclude that (co, ll . 114) is separable (only quote relevant results) (iii) Show that the closed unit ball in (a-II ·...
(5) Let S denote the set of sequences whose series are absolutely convergent. We define two...
(5) Let S denote the set of sequences whose series are absolutely convergent. We define two norms on S by ll{an} olla ΣΙΑ Janlı {an) oloup = sup{lano- no 1 (Note that S is the set of sequences such that a1 <. The sup-norm is sometimes called the o-norm.) Define a linear operator : S R by E({en}_o) - an NO (i) Compute the operator norm of using |-|- (ii) Show that the operator norm of using sup is unbounded.
3. (a) Let (R, τe) be the usual topology on R. Find the limit point set...
3. (a) Let (R, τe) be the usual topology on R. Find the limit point set of the following subsets of R (i) A = { n+1 n : n ∈ N} (ii) B = (0, 1] (iii) C = {x : x ∈ (0, 1), x is a rational number (b) Let X denote the indiscrete topology. Find the limit point set A 0 of any subset A of X. (c) Prove that a subset D of X is...
1. [4-+6+6-16 points Let /°0 denote the vector space of bounded sequences of real numbers, with a...
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...
9 Let co , a, and 〈æ be the Banach spaces consisting of all complex sequences x={ i-1, 2, 3,...,...
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...
Topology (c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup norm of C[o, 1]. (i) Show that 5 is closed under pointwise mu...
Topology
(c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup norm of C[o, 1]. (i) Show that 5 is closed under pointwise multiplication, that is,if f,0€万 then fg P and, moreover, llfglloo for all f,g E P
(c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup...
JUST PART C 10. Let X denote the subset of R consisting of all sequences (x1,x2,...)...
JUST PART C
10. Let X denote the subset of R consisting of all sequences (x1,x2,...) such that x converges. (You may assume the standard facts about infinite series. In case they are not familiar to you, we shall give them in Exercise 11 of the next section.) .(6,X) = (3x) (a) Show that if x, y e X, then xyil converges. [Hint: Use (b) of Exercise 9 nu er og to show that the partial sums are bounded.] 2uur...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Def...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
Please help me solve 3,4,5 3- For all n € N, let an = 1. Let...
Please help me solve 3,4,5
3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover...
Define where S is the collection of all real valued sequences i.e. S = {x : N → R} and we denote ...
Define where S is the collection of all real valued sequences i.e. S = {x : N → R} and we denote xi for the ith element a the sequence x E S. Take for any x EL (i) Show that lic 12 (where recall 1-(x є s i Izel < oo)) (ii) Is l? Prove this or find a counterexample to show that these two sets do not coinside (iii) ls e c loc where recall looー(x є sl...
Topology
(b) Let S denote the subset of co consisting of sequences with rational entries of which at most finitely many are nonzero. (i) Show that S is dense in co with the sup norm. [Hint: Show that for every r E co and every ε > 0, there exists y S such that llx-yI100 < ε.j (ii) Conclude that (co, ll . 114) is separable (only quote relevant results) (iii) Show that the closed unit ball in (a-II ·...
(5) Let S denote the set of sequences whose series are absolutely convergent. We define two norms on S by ll{an} olla ΣΙΑ Janlı {an) oloup = sup{lano- no 1 (Note that S is the set of sequences such that a1 <. The sup-norm is sometimes called the o-norm.) Define a linear operator : S R by E({en}_o) - an NO (i) Compute the operator norm of using |-|- (ii) Show that the operator norm of using sup is unbounded.
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...
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...
Topology
(c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup norm of C[o, 1]. (i) Show that 5 is closed under pointwise multiplication, that is,if f,0€万 then fg P and, moreover, llfglloo for all f,g E P
(c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup...
JUST PART C
10. Let X denote the subset of R consisting of all sequences (x1,x2,...) such that x converges. (You may assume the standard facts about infinite series. In case they are not familiar to you, we shall give them in Exercise 11 of the next section.) .(6,X) = (3x) (a) Show that if x, y e X, then xyil converges. [Hint: Use (b) of Exercise 9 nu er og to show that the partial sums are bounded.] 2uur...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
Please help me solve 3,4,5
3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover...
Define where S is the collection of all real valued sequences i.e. S = {x : N → R} and we denote xi for the ith element a the sequence x E S. Take for any x EL (i) Show that lic 12 (where recall 1-(x є s i Izel < oo)) (ii) Is l? Prove this or find a counterexample to show that these two sets do not coinside (iii) ls e c loc where recall looー(x є sl...
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