Due: Friday, March 30, 2007
Math 360 - Advanced Calculus / Problem Set 8 (two pages)
1) Let X, d be a metric space, and A⊆Xa non-empty subspace. Prove the following:
a) For x∈Xthe following are equivalent:
i) x∈A.
ii) ∀ > 0 one has: B(x, )∩A6=∅.
iii) ∃(xn)n,xn∈Asuch that xn→x.
b) Suppose that Xis complete. Then Ais complete ⇐⇒ Ais closed in X.
2) Recall the context from HW #6, Problem 8: We consider X0, d0and X00, d00 two metric spaces, and we
endow X:= X0×X00 with the metric d, which is either dmax or dsum. Prove the following:
a) Let (xn)nbe a sequence in X, hence xn= (x0
n, x00
n) with x0
n∈X0,x00
n∈X00. Then (xn)nis convergent
in X⇐⇒ (x0
n)nand (x00
n)nare convergent in X0, respectively X00.
b) Xis complete ⇐⇒ X0and X00 are complete.
3) Consider
R
nendowed with any of the metrics: dmax , the Euclidean distance d, or dsum. Let (xk)kbe a
sequence in
R
n, hence xk= (x1k, . . . , xnk), xj k ∈
R
for all k, 1 ≤j≤n(WHY?). Prove the following:
a) xk→x:= (x1, . . . , xn) in
R
n⇐⇒ xj k →xjin
R
for all j= 1, . . . , n.
b) (xk)kis Cauchy in
R
n⇐⇒ (xj k)kis Cauchy in
R
for all j= 1, . . . , n.
•Deduce from this: First,
R
nis complete. Second, A⊆
R
nis complete ⇐⇒ A=A.
4) Let A⊆
R
nbe a non-empty subspace. Prove the following:
a) Ais bounded ⇐⇒ Ais totally bounded.
b) Using the facts above and the characterization of compact metric spaces deduce the following fa-
mous/important fact:
Theorem (Heine–Borel). A non-empty subspace A⊂
R
nis compact ⇐⇒ Ais bounded and closed in
R
n.
5) Let kbe a base field. Let V,Wbe k-vector spaces, and f:V→Wak-linear map, i.e., a morphism of
k-vector spaces. Give complete proofs of the following facts mentioned in the class:
a) For λ∈kand v∈Vone has: λ·v= 0V⇐⇒ λ= 0kor v= 0V.
b) ker(f):={v∈V|f(v) = 0W}and im(f) := f(V) are k-vector subspaces of V, respectively W.
c) fis an isomorphism ⇐⇒ fis bijective, i.e., the inverse map of a bijective k-linear map is again
k-linear.
6) Give complete proofs/answers of the following:
a) The set of continuous bounded functions Cb(T,
R
) on a topological space Tis a vector space over
R
and
an
R
-algebra.
b) The set of polynomials
R
[X] in the variable Xis a vector space over
R
and an
R
-algebra.
c) For every q≥1, the spaces lq={(xn)n|xn∈
R
,Pn|xn|qis convergent}are vector spaces over
R
.
What is the relation between lqand lq0for q≤q0?
7) Answer the following:
a) For all n≥1, ∃injective
R
-linear maps
R
n→lq, and ∃injective
R
-linear maps
R
[X]→lq.
b) Are there surjective
R
-linear maps
R
n→l1, or
R
[X]→l1?
8) Show the following:
a)
C
is in a canonical way an algebra over
R
.
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