144 CHAPTER 5. FUNCTIONS ON NORMED LINEAR SPACES

Since A is compact, let U (y1) , · · · ,U (yl) cover A. Let

fx ≡max(

fxy1 , fxy2 , · · · , fxyl

).

Then fx ∈A and fx (z)> h(z)−ε for all z∈ A and fx (x) = h(x). This implies that for eachx ∈ A there exists an open set V (x) containing x such that for z ∈ V (x), fx (z) < h(z)+ ε.Let V (x1) , · · · ,V (xm) cover A and let f ≡ min( fx1 , · · · , fxm). Therefore, f (z) < h(z)+ ε

for all z ∈ A and since fx (z)> h(z)−ε for all z ∈ A, it follows f (z)> h(z)−ε also and so| f (z)−h(z)|< ε for all z. Since ε is arbitrary, this shows h ∈A and proves A =C (A;R).■

5.11 Connectedness in Normed Linear SpaceThe main result is that a ball in a normed linear space is connected. This is the nextlemma. From this, it follows that for an open set, it is connected if and only if it is arcwiseconnected.

Lemma 5.11.1 In a normed vector space, B(z,r) is arcwise connected.

Proof: This is easy from the convexity of the set. If x,y ∈ B(z,r) , then let γ (t) =x+ t (y−x) for t ∈ [0,1] .

∥x+ t (y−x)−z∥= ∥(1− t)(x−z)+ t (y−z)∥

≤ (1− t)∥x−z∥+ t ∥y−z∥< (1− t)r+ tr = r

showing γ (t) stays in B(z,r).■

Proposition 5.11.2 If X ̸= /0 is arcwise connected, then it is connected.

Proof: Let p ∈ X . Then by assumption, for any x ∈ X , there is an arc joining p and x.This arc is connected because it is the continuous image of an interval which is connected.Since x is arbitrary, every x is in a connected subset of X which contains p. Hence Cp = Xand so X is connected. ■

Theorem 5.11.3 Let U be an open subset of a normed vector space. Then U isarcwise connected if and only if U is connected. Also the connected components of an openset are open sets.

Proof: By Proposition 5.11.2 it is only necessary to verify that if U is connected andopen in the context of this theorem, then U is arcwise connected. Pick p ∈U . Say x ∈Usatisfies P if there exists a continuous function, γ : [a,b]→ U such that γ (a) = p andγ (b) = x.

A≡ {x ∈U such that x satisfies P .}

Ifx∈A, then Lemma 5.11.1 implies B(x,r)⊆U is arcwise connected for small enoughr. Thus letting y ∈ B(x,r) , there exist intervals, [a,b] and [c,d] and continuous functionshaving values in U , γ,η such that γ (a) = p,γ (b) = x,η (c) = x, and η (d) = y. Then letγ1 : [a,b+d− c]→U be defined as

γ1 (t)≡{γ (t) if t ∈ [a,b]η (t + c−b) if t ∈ [b,b+d− c]