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Exercise 29: Let FCK for some F closed and K compact. Use the definition of a...
1. (a) Prove that a closed subset of a compact set is compact. (b) Let a,...
1. (a) Prove that a closed subset of a compact set is compact. (b) Let a, b € R and f: R → R, x H ax + b. Prove that f is continuous. Is f uniformly continuous?
Use Alaoglu's theorem to show that if X is a Banach space, there is some compact...
Use Alaoglu's theorem to show that if X is a Banach space, there is some compact space Y such that X is isometrically isomorphic to a closed subspace of C(Y). HINT: What set is compact in Alaoglu's theorem?
Use Alaoglu's theorem to show that if X is a Banach space, there is some compact space Y such that X is isometrically isomorphic to a closed subspace of C(Y). HINT: What set is compact in Alaoglu's theorem?
hint This exercise 5 to use the definition of Riemann integral F. Let f : [a,...
hint
This exercise 5 to use the definition of Riemann integral
F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
5- Recall that a set KCR is said to be compact if every open cover for...
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...
1. Let k E C and define f:C+C to be the constant function f(x) = k....
1. Let k E C and define f:C+C to be the constant function f(x) = k. Use part (1) or (2) of to prove that f is continuous. (1) For every closed set FCC, the preimage f-1(F) is F' n D for some closed set F' (2) For every open set G C C, the preimage f-1(G) is G'' n D for some open set G'.
(2) Let X be a locally compact Hausdorff space, and let μ be a regular Borel measure on X such that μ(X) = +oo. Show that there is a non-negative function f CO(X) such that Jfdlı-+oo. Idea. Construct...
(2) Let X be a locally compact Hausdorff space, and let μ be a regular Borel measure on X such that μ(X) = +oo. Show that there is a non-negative function f CO(X) such that Jfdlı-+oo. Idea. Construct a sequence {K f-Σ001 nzfn, n} of disjoint compact sets K n with μ(An) > n and set where fn E Co(X) with XKn S f 31 く!
(2) Let X be a locally compact Hausdorff space, and let μ be a...
* Exercise 10. Let M be a (non-empty) compact metric space and f: M → M...
* Exercise 10. Let M be a (non-empty) compact metric space and f: M → M a continuous map such that for every ε > 0 there exists x E M such that d(f(x), z) < E. Show that there exists y M such that f()y Hint: consider the map g: MR defined by g(x)=d(f(x),z).] [8 marks]
plz use the definition solve the question Definition 1. Given a set A CR, an elementu...
plz use the definition solve the question
Definition 1. Given a set A CR, an elementu ER is an interior point of A if there exists an e > 0 such that (x - 5,3 +E) CA. The interior of A is the set Aº consisting of all interior points of A. A set A is called open if A= A'. Definition 2. Given a set A CR, an element X ER is a limit point of A if for...
Real Analysis II Please do it without using Heine-Borel's theorem and do it only if you're sure Problem: Let E be a closed bounded subset of En and r be any function mapping E to (0,∞). Then t...
Real Analysis II
Please do it without using Heine-Borel's theorem
and do it only if you're sure
Problem: Let E be a closed bounded subset of
En and r be any function mapping E to
(0,∞). Then there exists finitely many points yi ∈ E, i
= 1,...,N such that
Here Br(yi)(yi) is the open ball
(neighborhood) of radius r(yi) centered at
yi.
Also, following definitions & theorems should help
that
E CUBy Definition. A subset S of a topological...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that t...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
1. (a) Prove that a closed subset of a compact set is compact. (b) Let a, b € R and f: R → R, x H ax + b. Prove that f is continuous. Is f uniformly continuous?
Use Alaoglu's theorem to show that if X is a Banach space, there is some compact space Y such that X is isometrically isomorphic to a closed subspace of C(Y). HINT: What set is compact in Alaoglu's theorem?
Use Alaoglu's theorem to show that if X is a Banach space, there is some compact space Y such that X is isometrically isomorphic to a closed subspace of C(Y). HINT: What set is compact in Alaoglu's theorem?
hint
This exercise 5 to use the definition of Riemann integral
F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...
1. Let k E C and define f:C+C to be the constant function f(x) = k. Use part (1) or (2) of to prove that f is continuous. (1) For every closed set FCC, the preimage f-1(F) is F' n D for some closed set F' (2) For every open set G C C, the preimage f-1(G) is G'' n D for some open set G'.
(2) Let X be a locally compact Hausdorff space, and let μ be a regular Borel measure on X such that μ(X) = +oo. Show that there is a non-negative function f CO(X) such that Jfdlı-+oo. Idea. Construct a sequence {K f-Σ001 nzfn, n} of disjoint compact sets K n with μ(An) > n and set where fn E Co(X) with XKn S f 31 く!
(2) Let X be a locally compact Hausdorff space, and let μ be a...
* Exercise 10. Let M be a (non-empty) compact metric space and f: M → M a continuous map such that for every ε > 0 there exists x E M such that d(f(x), z) < E. Show that there exists y M such that f()y Hint: consider the map g: MR defined by g(x)=d(f(x),z).] [8 marks]
plz use the definition solve the question
Definition 1. Given a set A CR, an elementu ER is an interior point of A if there exists an e > 0 such that (x - 5,3 +E) CA. The interior of A is the set Aº consisting of all interior points of A. A set A is called open if A= A'. Definition 2. Given a set A CR, an element X ER is a limit point of A if for...
Real Analysis II
Please do it without using Heine-Borel's theorem
and do it only if you're sure
Problem: Let E be a closed bounded subset of
En and r be any function mapping E to
(0,∞). Then there exists finitely many points yi ∈ E, i
= 1,...,N such that
Here Br(yi)(yi) is the open ball
(neighborhood) of radius r(yi) centered at
yi.
Also, following definitions & theorems should help
that
E CUBy Definition. A subset S of a topological...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
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