If Y is locally compact Hausdorff space ,prove that there is a
homomorphism C(XY,Z)
C(X,C(Y,Z))
and define the homomorphism.
If Y is locally compact Hausdorff space ,prove that there is a homomorphism C(XY,Z)C(X,C(Y,Z)) an...
(3) Let X be a locally compact Hausdorff space, let A be a ơ-algebra on X that includes B(X) and let μ be a regular measure on (Χ.Α). Show that the completion of μ is regular.
(3) Let X be a locally compact Hausdorff space, let A be a ơ-algebra on X that includes B(X) and let μ be a regular measure on (Χ.Α). Show that the completion of μ is regular.
(2) Let X be a locally compact Hausdorff space, and let μ be a regular Borel measure on X such that μ(X) = +oo. Show that there is a non-negative function f CO(X) such that Jfdlı-+oo. Idea. Construct a sequence {K f-Σ001 nzfn, n} of disjoint compact sets K n with μ(An) > n and set where fn E Co(X) with XKn S f 31 く!
(2) Let X be a locally compact Hausdorff space, and let μ be a...
Let
be a metric space and let
be the topology on
induced by
, and let
be a compact space. Prove that
is compact.
(x, d) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageAj,i=1,2,... na1 An
Please show the steps thoroughly, thank you so much!
Let X be a compact Hausdorff space and Y a subset of X. Let J/ be the ideal of functions in C(X) vanishing on Y. In general, Amay not be isomorphic to C(Y). Evaluate the following statement: If Y is closed in X, then Ais isomorphic to C(Y). If you answer true, is this isomorphism also isometric?
Let X be a compact Hausdorff space and Y a subset of X. Let...
Define φ : Q[x] → Q by φ()
=
.
(a) Prove that φ is a ring homomorphism.
(b) Find the kernel of φ.
and" + ...a12 + ao We were unable to transcribe this image
Let S2 denote the
2-dimensional sphere. Define the complex projective line
1 as the quotient space
2 \ {0} / ∼ , where ∼ is the equivalence relation on
2 \ {0} that x ∼ y if x = λy for some λ∈C. Prove that
S2 and
1 are homeomorphic.
Let S denote the 2-dimensional sphere. Define the complex projective line CP as the quotient space C {0}/~, where is the equivalence relation on {0} that I ~y if r...
We define the ring homomorphism
by
a) Show that the kernel of
is <x3 -2>, and that the image is
b) Conclude that
is a subfield of
SOLVE B only please
V : Q2 +R vf(x) = f[V2 We were unable to transcribe this imageQ(72) = a +672 +c72* a, b, c € 0 Q(2) We were unable to transcribe this image
A metric space (X, d) is totally bounded if, given
ε>0, there exists a finite subset =
of X, called an ε-net, such that for each x∈X there
exists
∈
such that d(x,)
< ε. Prove that if Y is a subset of a totally bounded space X
then, given ε>0, the subset Y has an ε-net and
therefore Y is also totally bounded.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to...
Let X and Y be a first countable spaces. Prove that f:XY
is continuous if whenever xnx
in X then f(xn
)f(x)
in Y
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let
be an inner product space (over
or
), and
. Prove that
is an eigenvalue of
if and only if
(the conjugate of
) is an eigenvalue of
(the adjoint of
).
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageTEL(V) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to...