Download University of Bergen General Functional Analysis Problems 4 1) Let

University of Bergen
General Functional Analysis
Problems 4
1) Let X be a metric space. Let {fn }∞
n=1 be a sequence of continuous functions from X to a metric space
Y that converges to a function f uniformly on each compact subset K of X. Then f is continuous.
2) Prove the Ascoli-Arzelá theorem in the following form. Let X be a compact metric space, and
let F be a family of real-valued continuous functions defined on X. Then F is compact
if and only if F is closed, uniformly bounded, and equicontinuous.
3) Using the definition of a closed set, closure E, interior E ◦ of a set E in a topological space (X, τ ),
prove the following properties
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
E ∈ E,
E = E,
(A ∪ B) = A ∪ B,
a set F is closed if and only if F = F ,
E is the set of points of closure of E,
E ◦ ⊂ E,
E ◦◦ = E ◦ ,
(A ∩ B)◦ = A◦ ∩ B ◦ ,
E ◦ is the set of interior points of E,
(X\)◦ = X \ E
∞
4) Let {xn }∞
n=1 be a sequence in a topological space. If {xn }n=1 has a subsequence converging to x, then
∞
x is a cluster (limit) point of {xn }n=1 . Give an example showing that converse of this statement is not
true in an arbitrary topological space.
5) Let X be a nonempty set of points and let A be any collection of subsets of X. Then there is a weakest
topology τ that contains A.
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