Let {fn} ⊆ C([0, 1]). Prove that {fn} converges uniformly on [0, 1] if and only if {fn} is equicontinuous on [0, 1] and converges pointwise on [0, 1].
Let {fn} ⊆ C([0, 1]). Prove that {fn} converges uniformly on [0, 1] if and only if {fn} is equicontinuous on [0, 1] and...
(5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded and equicontinuous function f E a (5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded and equicontinuous function f E a
Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges uniformly to 0. (b) Show that .., XER, NEN. converges pointwise to a function discontinuous at the origin.
For n ∈ N let fn : [0, 1] → R be given by fn(x) = sin((1 + n)x) / ( 1 + n ) ^(1/2) . Prove that {fn} is equicontinuous on [0, 1].
3. For each n E N let fn : (1, 0) -+ R be given by f/(x) = Find the function f : (1, 0) - R to which {fn} converges pointwise. Prove that the convergence is not uniform 3. For each n E N let fn : (1, 0) -+ R be given by f/(x) = Find the function f : (1, 0) - R to which {fn} converges pointwise. Prove that the convergence is not uniform
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
(a) Let Sk: KCR + R" be an equicontinuous sequence of functions on a compact set K converging pointwise. Prove that the convergence is uniform. (a) Use (a) to obtain that fn(x) x2 x2 +(1-nx)2, 0 < x < 1 is not equicontinuous. =
(Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly) 0. Prove that fn → f pointwise but (Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly) 0. Prove that fn → f pointwise but
Let f(3) = 1 (a) Prove {f} 1 + nx converges to 0 pointwise on (-0,00). (b) Prove or disprove {n} , converges to 0 uniformly on (-0, 0);
Tamo . Suppose that a sequence of functions fn converges pointwise to a function f on a set E, but there exists a sequence of points In E E such that \fn(2n) – f(2n) > for some strictly positive l. Then fn does not converge uniformly to f on E. (You don't need to prove this here, but it should be clear why this is true.) Now let nar2 fn(L) = 2 +n323 Show that fn converges pointwise on [0,0]...