516 CHAPTER 18. TOPOLOGICAL VECTOR SPACES

still denoted as {xn} which converges weakly to x ∈ B(0,r). Then Axn → y and xn → xweakly. Thus (x,y) is in the weak closure of the graph of A,

{(x,Ax) : x ∈ Xi}

This set is strongly closed and convex and hence it is weakly closed by Theorem 18.3.3 soy = Ax and this shows A

(B(0,r)

)is closed. In the other case where Xi is the dual space of

a separable Banach space, it follows from Corollary 17.5.6 there exists a subsequence stilldenoted as {xn} such that xn→ x weak ∗ and similarly, (x,y) is in the weak ∗ closure of thegraph of A which shows again by Theorem 18.3.3 that (x,y) is in the graph of A, showingagain that A

(B(0,r)

)is closed.

Suppose 18.7.31. Then letting y∗ ∈ Y ′,

||A∗1y∗|| = sup||x1||X1

≤1|y∗ (A1x1)|

≤ sup||x2||X2

≤k|y∗ (A2x2)|= k ||A∗2y∗||

which shows 18.7.32.Now suppose 18.7.32. Then if 18.7.31 does not hold, it follows from the first part which

gives Ai

(B(0,r)

)a closed set, there exists

A1x0 ∈ A1

(B(0,1)

)\A2

(B(0,k)

)Now A2

(B(0,k)

)is closed and convex, hence weakly closed, and so by Theorem 18.2.5

there exists y∗0 ∈ Y ′ such that

Rey∗0(

A2

(B(0,k)

))< c < Rey∗0 (A1x0)

and so

||A∗1y∗0|| = sup||x1||X1

≤1|y∗0 (Ax1)| ≥ Rey∗0 (A1x0)

> c > Rey∗0 (A2 (x2)) = ReA∗2y∗0 (x2)

whenever x2 ∈ B(0,k) and so, taking the supremum of all such x2,

||A∗1y∗0||> c > k ||A∗2y∗0|| ,

contradicting 18.7.32.