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Q3. lut (H, J) be a compart metric space . Suppose ful is sequence of continuous...
Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R. Suppose t...
Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R. Suppose that, for each the sequence (fe(x))ke N 1s a monotonic sequence which converges to (x). Show that r є X, k)kEN Converges to j uniformly.
Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R....
Please answer c d e 3. This problem shows that the metric space of continuous real-valued...
Please answer c d e
3. This problem shows that the metric space of continuous real-valued functions C([0, 1]) on the interval [0, 1is complete. Recall that we use the sup metric on C([0,1), so that df, 9) = sup{f (2) - 9(2): € (0,1]} (a) Suppose that {n} is a Cauchy sequence in C([0,1]). Show that for each a in 0,1], {Sn(a)} is a Cauchy sequence of real numbers. (b) Show that the sequence {fn(a)} converges. We define f(a)...
A subset D of a metric space (X, d) is dense if every member of X...
A subset D of a metric space (X, d) is dense if every member of
X is a limit of a sequence of elements from D.
Suppose (X,d) and (Y,ρ) are metric spaces and D is a dense
subset of X.
1. Prove that if f : D -» Y is uniformly continuous then there exists an extension15 of f to a if dn (E D) e X define 7(x) lim f(d,) uniformly continuous function f:X * Y. Hint: 2....
3. Suppose X is a metric space with a sequence of points Xn e X with...
3. Suppose X is a metric space with a sequence of points Xn e X with the property that for each n + m we have d(Xn, Xm) = 1. Prove that no subsequence of xn converges, and that therefore X is not compact. Hint: You could use the previous problem.
Let X be metric space, and let g:X + R be uniformly continuous and h: R+R...
Let X be metric space, and let g:X + R be uniformly continuous and h: R+R be continuous. (c) If h is uniformly continuous on R, then goh is uniformly continuous. (b) If g(x) = {g(x) : x € X} is a bounded set,(i.e. there exists M > 0 such that g(x) < M for all X E X.) then go h is uniformly continuous.
Jet f be continuons one to one m compact metric space X onto a metric space...
Jet f be continuons one to one m compact metric space X onto a metric space Y. Prove that f'Y ~ X is continuoms (Hint: use this let X and Y e metric space, and let f be function from X to Y which is one to one and onto then the following three statments are equivalent. frs open, f is closed, f is continuous.
10 Let fn be a sequence of functions that converges uniformly to f on a set E and satisfies IfGİ M for all 1,2 and all r e E. Suppose g is a continuous function on [-MI, M]. Show that g(Um(x)) un...
10 Let fn be a sequence of functions that converges uniformly to f on a set E and satisfies IfGİ M for all 1,2 and all r e E. Suppose g is a continuous function on [-MI, M]. Show that g(Um(x)) uniformly to g(f(r)) on E
10 Let fn be a sequence of functions that converges uniformly to f on a set E and satisfies IfGİ M for all 1,2 and all r e E. Suppose g is a continuous...
i) Does Lebesgue lemma hold true in the plane? Justify your answer! ii) Let (X, d1) be a compact metric space and (Y, d2) a metric space. Suppose that f : X → Y is continuous. Use Lebesgue lemma to sh...
i) Does Lebesgue lemma hold true in the plane? Justify your answer! ii) Let (X, d1) be a compact metric space and (Y, d2) a metric space. Suppose that f : X → Y is continuous. Use Lebesgue lemma to show that for every > 0 there exists δ > 0 such that if d1(x, y) < δ then d2(f(x), f(y)) < , that is, f is uniformly continuous.
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...
Let E ç R. Suppose that (felre is a sequence of continuous function from E to...
Let E ç R. Suppose that (felre is a sequence of continuous function from E to R 120ml o prove that if form of uniformly on Eta then fis continuous an E. e Giva an example where for f almost uniformly on E but f is not continuons on E
Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R. Suppose that, for each the sequence (fe(x))ke N 1s a monotonic sequence which converges to (x). Show that r є X, k)kEN Converges to j uniformly.
Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R....
Please answer c d e
3. This problem shows that the metric space of continuous real-valued functions C([0, 1]) on the interval [0, 1is complete. Recall that we use the sup metric on C([0,1), so that df, 9) = sup{f (2) - 9(2): € (0,1]} (a) Suppose that {n} is a Cauchy sequence in C([0,1]). Show that for each a in 0,1], {Sn(a)} is a Cauchy sequence of real numbers. (b) Show that the sequence {fn(a)} converges. We define f(a)...
A subset D of a metric space (X, d) is dense if every member of
X is a limit of a sequence of elements from D.
Suppose (X,d) and (Y,ρ) are metric spaces and D is a dense
subset of X.
1. Prove that if f : D -» Y is uniformly continuous then there exists an extension15 of f to a if dn (E D) e X define 7(x) lim f(d,) uniformly continuous function f:X * Y. Hint: 2....
3. Suppose X is a metric space with a sequence of points Xn e X with the property that for each n + m we have d(Xn, Xm) = 1. Prove that no subsequence of xn converges, and that therefore X is not compact. Hint: You could use the previous problem.
Let X be metric space, and let g:X + R be uniformly continuous and h: R+R be continuous. (c) If h is uniformly continuous on R, then goh is uniformly continuous. (b) If g(x) = {g(x) : x € X} is a bounded set,(i.e. there exists M > 0 such that g(x) < M for all X E X.) then go h is uniformly continuous.
Jet f be continuons one to one m compact metric space X onto a metric space Y. Prove that f'Y ~ X is continuoms (Hint: use this let X and Y e metric space, and let f be function from X to Y which is one to one and onto then the following three statments are equivalent. frs open, f is closed, f is continuous.
10 Let fn be a sequence of functions that converges uniformly to f on a set E and satisfies IfGİ M for all 1,2 and all r e E. Suppose g is a continuous function on [-MI, M]. Show that g(Um(x)) uniformly to g(f(r)) on E
10 Let fn be a sequence of functions that converges uniformly to f on a set E and satisfies IfGİ M for all 1,2 and all r e E. Suppose g is a continuous...
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...
Let E ç R. Suppose that (felre is a sequence of continuous function from E to R 120ml o prove that if form of uniformly on Eta then fis continuous an E. e Giva an example where for f almost uniformly on E but f is not continuons on E
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