Question

Show that a weakly open subset U of a Banach space is also open in the norm topology.

Answer #1

Show that an open ball relative to the usual Euclidean distance
is a subset of an open ball relative to the infinity norm distance
and vice versa.

Prove that if T is a discrete topological space (i.e., every subset
of T is open) and ∼ is any equivalence relation on T , then the
quotient space T / ∼ is also discrete topological space. [Hint: It
is a very short and straightforward proof.]

1. Consider the set U={(x,y) in R2| -1<x<1 and y=0}. Is U
open in R2? Is it open in R1? Is it open as a subspace of the disk
D={(x,y) in R2 | x^2+y^2<1} ?
2. Is there any subset of the plane in which a single point set
is open in the subspace topology?

a)Suppose U is a nonempty subset of the vector space V over
field F. Prove that U is a subspace if and only if cv + w ∈ U for
any c ∈ F and any v, w ∈ U
b)Give an example to show that the union of two subspaces of V
is not necessarily a subspace.

show that each subset of R is not compact by describing an open
cover for it that has no finite subcover . [1,3) , also explain a
little bit of finite subcover, what does it mean like a finite.

Show that a compact subset of a Hausdorff space is closed in
detail.

Show that if a subset S has a maximum, then the maximum is also
the supremum. Similarly, show that if S has a minimum, then the
minimum is also the infimum

1. V is a subspace of inner-product space R3,
generated by vector
u =[2 2 1]T and v
=[ 3 2 2]T.
(a) Find its orthogonal complement space V┴ ;
(b) Find the dimension of space W = V+ V┴;
(c) Find the angle θ between u and
v and also the angle β between u
and normalized x with respect to its 2-norm.
(d) Considering v’ =
av, a is a scaler, show the
angle θ’ between u and...

Let T be the half-open interval topology for R, defined in
Exercise 4.6.
Show that (R,T) is a T4 - space.
Exercise 4.6
The intersection of two half-open intervals of the form [a,b) is
either empty or a half-open interval. Thus the family of all unions
of half-open intervals together with the empty set is closed under
finite intersections, hence forms a topology, which has the
half-open intervals as a base.

Real Topology: let A={1/n : n is natural} be a subset of the real
numbers. Is A open closed, or neither? Justify your answer.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 38 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 3 hours ago

asked 3 hours ago