Question

Suppose K is a nonempty compact subset of a metric space X and x∈X.

**Show, there is a nearest point p∈K to x; that is, there
is a point p∈K such that, for all other q∈K,
d(p,x)≤d(q,x).**

[Suggestion: As a start, let S={d(x,y):y∈K} and show there is a sequence (qn) from K such that the numerical sequence (d(x,qn)) converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}.

**Show, there is a point z∈X and distinct points a,b∈T
that are nearest points to z. Let X=(0,∞)∪{−1} (as a subspace of
R). Show there is no nearest point to −1 from the closed (in X) set
K=(0,1].**

Answer #1

Problem 6. For a closed convex nonempty subset
K of a Hilbert space H and x ∈ H, denote by P x ∈ K a unique
closest point to x among points in K, i.e. P x ∈ K such that
||P x − x|| ≤ ||y − x||, for all y ∈ K.
First show that such point P x exists and unique. Next prove
that all x, y ∈ H
||P x − P y|| ≤ ||x −...

Let (X,d)
be a complete
metric space, and T
a d-contraction
on X,
i.e., T:
X
→ X
and there exists a q∈
(0,1) such that for all x,y
∈ X,
we have d(T(x),T(y))
≤ q∙d(x,y).
Let a
∈ X,
and define a sequence (xn)n∈Nin
X
by
x1 :=
a
and ∀n ∈
N: xn+1
:= T(xn).
Prove, for all n
∈ N,
that d(xn,xn+1)
≤ qn-1∙d(x1,x2).
(Use
the Principle of Mathematical Induction.)
Prove that (xn)n∈N
is a d-Cauchy
sequence in...

Let (X, d) be a compact metric space and let A ⊆ X. Suppose that
A is not compact. Prove that there exists a continuous function f :
A → R, from (A, d) to (R, d|·|), which is not uniformly
continuous.

(2)
If K is a subset of (X,d), show that K is compact if and only if
every cover of K by relatively open subsets of K has a finite
subcover.

Let (X, d) be a compact metric space and F: X--> X be a
function such that d(F(x), F(y)) < d(x, y). Let G: X --> R be
a function such that G(x) = d(F(x), x). Prove G is continuous
(assume that it is proved that F is continuous).

Suppose (an), a sequence in a metric space X, converges to L ∈
X. Show, if σ : N → N is one-one, then the sequence (bn = aσ(n))n
also converges to L.

Let (X,d) be a metric space which contains an infinite countable
set Ewith the property x,y ∈ E ⇒ d(x,y) = 1.
(a) Show E is a closed and bounded subset of X. (b) Show E is
not compact.
(c) Explain why E cannot be a subset of Rn for any n.

Let X, Y be metric spaces, with Y complete. Let S ⊂ X and let f
: S → Y be uniformly continuous. (a) Suppose p ∈ S closure and (pn)
is a sequence in S with pn → p. Show that (f(pn)) converges in y to
some point yp.

Each of the following deﬁnes a metric space X which is a subset
of R^2 with the Euclidean metric, together with a subset E ⊂ X. For
each,
1. Find all interior points of E,
2. Find all limit points of E,
3. Is E is open relative to X?,
4. E is closed relative to X?
I don't worry about proofs just answers is fine!
a) X = R^2, E = {(x,y) ∈R^2 : x^2 + y^2 = 1,...

is about metric spaces:
Let X be a metric discret space show that a sequence x_n in X
converge to l in X iff x_n is constant exept for a finite number of
points.

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