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...
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x E A, we have lÍs(x)-f(x)|く for all n > N. Then f is uniformly continuous. Note: We could say that the "sequence of functions" f "converges to the function" f. These are not defined terms for...
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.
Q3. lut (H, J) be a compart metric space . Suppose ful is sequence of continuous function fo: M-IR which converges uniformly to f: MIR Prove that the sequence 4 Converges uniformly to f².
I need help proving this theorem Theorem 5.6.3 Let (fk)ken be a sequence of Riemann integrable functions defined on a closed generalized rectangle R CR”. Suppose that (fk)ken converges uniformly to a function f on R. Then f is Riemann integrable on R, and Sef = limk+ SR fk.
(a) Suppose K is a compact subset of a metric space (X, d) and x є X but x K Show that there exist disjoint, open subsets of Gi and G2 of (X, d) such that r E Gi and KG2. (Hint: Use the version of compactness we called "having a compact topology." You will also need the Hausdorff property.) b) Now suppose that Ki and K2 are two compact, disjoint subsets of a metric space (X, d). Use (a)...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
Let (X, d) be a compact metric space. Prove that if F ⊆ C(X) is equicontinuous then it is uniformly equicontinuous.
Let (X, dx), (Y, dy) be metric spaces and fn be a sequence of functions fn: XY Prove that if {fn} converges uniformly on X then for any a є x lim lim fn()- lim lim /) xa n-00 Let (X, dx), (Y, dy) be metric spaces and fn be a sequence of functions fn: XY Prove that if {fn} converges uniformly on X then for any a є x lim lim fn()- lim lim /) xa n-00
3. Suppose that (M, ρ) is a compact metric space and f : (M, p)-+ (M,p) is a function such that (Vz, y E M) ρ (z, y) ρ (f (x), f (y)). a. Let x E (M, ρ) and consider the sequence of points {f(n) (X)}n 1 . (Remember: fn) denotes the composition of f with itself, n times, so for each n, f+() rn, k E N) such that ρ (f(m) (x) ,f(n +k) (r)) < ε ....
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.