Multivariable Advanced Calculus

4.6. EXERCISES 8316. A normed vector space, V is separable if there is a countable set {w k } ∞ k=1 suchthat whenever B (x, δ) is an open ball in V, there exists some w k in this open ball.Show that F n is separable. This set of points is called a countable dense set.17. Let V be any normed vector space with norm ||·||. Using Problem 13 show thatV is separable.18. Suppose V is a normed vector space. Show there exists a countable set of openballs B ≡ {B (x k , r k )} ∞ k=1having the remarkable property that any open set, U isthe union of some subset of B. This collection of balls is called a countable basis.Hint: Use Problem 17 to get a countable dense dense set of points, {x k } ∞ k=1 andthen consider balls of the form B ( )x k , 1 r where r ∈ N. Show this collection ofballs is countable and then show it has the remarkable property mentioned.19. Suppose S is any nonempty set in V a finite dimensional normed vector space.Suppose C is a set of open sets such that ∪C ⊇ S. (Such a collection of sets iscalled an open cover.) Show using Problem 18 that there are countably manysets from C, {U k } ∞ k=1 such that S ⊆ ∪∞ k=1 U k. This is called the Lindeloff propertywhen every open cover can be reduced to a countable sub cover.20. A set, H in a normed vector space is said to be compact if whenever C is a set ofopen sets such that ∪C ⊇ H, there are finitely many sets of C, {U 1 , · · · , U p } suchthatH ⊆ ∪ p i=1 U i.Show using Problem 19 that if a set in a normed vector space is sequentiallycompact, then it must be compact. Next show using Problem 14 that a set in anormed vector space is compact if and only if it is closed and bounded. Explainwhy the sets which are compact, closed and bounded, and sequentially compactare the same sets in any finite dimensional normed vector space