104 CHAPTER 4. LINEAR SPACES

Proof: In Lemma 4.4.7, let δ ,∆ go with ∥·∥ and δ̂ , ∆̂ go with ∥·∥1. Then using theinequalities of this lemma,

∥v∥ ≤ ∆ |θv| ≤ ∆

δ̂∥v∥1 ≤

∆∆̂

δ̂|θv| ≤ ∆

δ

∆̂

δ̂∥v∥

and so δ̂

∆∥v∥ ≤ ∥v∥1 ≤ ∆̂

δ∥v∥. Thus the norms are equivalent. ■

It follows right away that the closed and open sets are the same with two differentnorms. Also, all considerations involving limits are unchanged from one norm to another.

Corollary 4.4.10 Consider the metric spaces (V,∥·∥1) ,(V,∥·∥2) where V has dimen-sion n. Then a set is closed or open in one of these if and only if it is respectively closed oropen in the other. In other words, the two metric spaces have exactly the same open andclosed sets. Also, a set is bounded in one metric space if and only if it is bounded in theother.

Proof: This follows from Theorem 3.6.2, the theorem about the equivalent formulationsof continuity. Using this theorem, it follows from Theorem 4.4.9 that the identity mapI (x)≡x is continuous. The reason for this is that the inequality of this theorem implies thatif ∥vm−v∥1→ 0 then ∥Ivm− Iv∥2 = ∥I (vm−v)∥2→ 0 and the same holds on switching1 and 2 in what was just written.

Therefore, the identity map takes open sets to open sets and closed sets to closed sets.In other words, the two metric spaces have the same open sets and the same closed sets.

Suppose S is bounded in (V,∥·∥1). This means it is contained in B(0,r)1 where thesubscript of 1 indicates the norm is ∥·∥1 . Let δ ∥·∥1 ≤ ∥·∥2 ≤ ∆∥·∥1 as described above.Then S⊆ B(0,r)1 ⊆ B(0,∆r)2 so S is also bounded in (V,∥·∥2). Similarly, if S is boundedin ∥·∥2 then it is bounded in ∥·∥1. ■

One can show that in the case of R where it makes sense to consider sup and inf, con-vergence of Cauchy sequences can be shown to imply the other definition of completenessinvolving sup, and inf.

4.5 Vitali Covering TheoremThese covering theorems make sense on any finite dimensional normed linear space. Thereare two which are commonly used, the Vitali theorem and the Besicovitch theorem. Thefirst adjusts the size of balls and the second does not. The Vitali theorem is the only one Iwill use in this book. See my larger book “Real and Abstract Analysis” for the Besicovitchtheorem.

The Vitali covering theorem is a profound result about coverings of a set in (X ,∥·∥)with balls. Usually we are interested in Rp with some norm. We will tacitly assume allballs have positive radius. They will not be single points. Before beginning the proof, hereis a useful lemma.

Lemma 4.5.1 In a normed linear space, B(x,r) = {y : ∥y−x∥ ≤ r} .

Proof: It is clear that B(x,r) ⊆ {y : ∥y−x∥ ≤ r} because if y ∈ B(x,r), then thereexists a sequence of points of B(x,r) ,{xn} such that ∥xn−y∥ → 0,∥xn∥ < r. However,this requires that ∥xn∥ → ∥y∥ and so ∥y∥ ≤ r. Now let y be in the right side. It sufficesto consider ∥y−x∥ = 1. Then you could consider for t ∈ (0,1) , x+ t (y−x) = z (t).