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1. If A CB CM, and if B is totally bounded, show that A is totally...
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 ·...
2. Show that the closed ball of radius 1 centered at 0 in Loo cannot be covered Hint: Think about...
2. Show that the closed ball of radius 1 centered at 0 in Loo cannot be covered Hint: Think about the result by finitely many closed balls of radius of Problem 14 in Section 7.1.)
2. Show that the closed ball of radius 1 centered at 0 in Loo cannot be covered Hint: Think about the result by finitely many closed balls of radius of Problem 14 in Section 7.1.)
1. Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint: Use an energy argument-multiply the ODE by y and integrate from r 0 to r R; use integration by parts and use th...
1. Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint: Use an energy argument-multiply the ODE by y and integrate from r 0 to r R; use integration by parts and use the boundedness at r0 to get the boundary term to vanish. (4.75) which is Bessel's equation. Condition (4.72) leads to the boundary condition y(R)0, (4.76) and we impose the boundedness requirement y(0) bounded (4.77)
1. Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint:...
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 for every r E co and every e >0, there exists y E S such that llr- yllle (ii) Conclude that (co, l is separable (only quote relevant results) (ii) Show that the closed unit ball in (coIl lis not compact. [Hint:...
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...
(a) Suppose f is continuously differentiable on the closed and bounded interval I = [0, 1]....
(a) Suppose f is continuously differentiable on the closed and bounded interval I = [0, 1]. Show that f is uniformly continuous on I. (b) Suppose g is continuously differentiable on the open interval J = (0,1). Give and example of such a function which is NOT uniformly continuous on J, and prove your answer.
Let f, (x) := lxl1+1/n, Π ε N, and f(x) 비파 Show Exercise 13: a) fn-f uniformly on all bounded intervals (a, b) C R. b) fn -f is not uniformly on all of R. Let f, (x) := lxl1+1/n, Π ε N, and f(...
Let f, (x) := lxl1+1/n, Π ε N, and f(x) 비파 Show Exercise 13: a) fn-f uniformly on all bounded intervals (a, b) C R. b) fn -f is not uniformly on all of R.
Let f, (x) := lxl1+1/n, Π ε N, and f(x) 비파 Show Exercise 13: a) fn-f uniformly on all bounded intervals (a, b) C R. b) fn -f is not uniformly on all of R.
(a) Show that for > 0, the first integral equals: 0 n-0 (b) Show that 1) n=0 defines a continuous function on Ω :-R\{0,-1,-2,-3, . . . } [Hint: consider the domain R\ U000 (-n-e,-n + ε) for any sm...
(a) Show that for > 0, the first integral equals: 0 n-0 (b) Show that 1) n=0 defines a continuous function on Ω :-R\{0,-1,-2,-3, . . . } [Hint: consider the domain R\ U000 (-n-e,-n + ε) for any small ε > 0.]
(a) Show that for > 0, the first integral equals: 0 n-0 (b) Show that 1) n=0 defines a continuous function on Ω :-R\{0,-1,-2,-3, . . . } [Hint: consider the domain R\ U000 (-n-e,-n + ε)...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx)...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...
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...
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 ·...
2. Show that the closed ball of radius 1 centered at 0 in Loo cannot be covered Hint: Think about the result by finitely many closed balls of radius of Problem 14 in Section 7.1.)
2. Show that the closed ball of radius 1 centered at 0 in Loo cannot be covered Hint: Think about the result by finitely many closed balls of radius of Problem 14 in Section 7.1.)
1. Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint: Use an energy argument-multiply the ODE by y and integrate from r 0 to r R; use integration by parts and use the boundedness at r0 to get the boundary term to vanish. (4.75) which is Bessel's equation. Condition (4.72) leads to the boundary condition y(R)0, (4.76) and we impose the boundedness requirement y(0) bounded (4.77)
1. Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint:...
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 e >0, there exists y E S such that llr- yllle (ii) Conclude that (co, l is separable (only quote relevant results) (ii) Show that the closed unit ball in (coIl lis not compact. [Hint:...
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...
(a) Suppose f is continuously differentiable on the closed and bounded interval I = [0, 1]. Show that f is uniformly continuous on I. (b) Suppose g is continuously differentiable on the open interval J = (0,1). Give and example of such a function which is NOT uniformly continuous on J, and prove your answer.
Let f, (x) := lxl1+1/n, Π ε N, and f(x) 비파 Show Exercise 13: a) fn-f uniformly on all bounded intervals (a, b) C R. b) fn -f is not uniformly on all of R.
Let f, (x) := lxl1+1/n, Π ε N, and f(x) 비파 Show Exercise 13: a) fn-f uniformly on all bounded intervals (a, b) C R. b) fn -f is not uniformly on all of R.
(a) Show that for > 0, the first integral equals: 0 n-0 (b) Show that 1) n=0 defines a continuous function on Ω :-R\{0,-1,-2,-3, . . . } [Hint: consider the domain R\ U000 (-n-e,-n + ε) for any small ε > 0.]
(a) Show that for > 0, the first integral equals: 0 n-0 (b) Show that 1) n=0 defines a continuous function on Ω :-R\{0,-1,-2,-3, . . . } [Hint: consider the domain R\ U000 (-n-e,-n + ε)...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...
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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