Separation Theorems and Krein-Milman’s Theorem

We start with the following interesting result: 8.1 Lemma

Let X be a TVS and 0 ≠ f ∈ X^*. Then f(G) is open in K whenever G is open in X.

Proof.

One can assume that G is non-empty and that 0 ≠ x₀ ∈ X is such that f(x₀) = 1. For any a ∈ G, it is required to show that f(a) is an interior point of f(G). Indeed, since G — a is an open 0—neighbourhood, it absorbs x₀; namely, there exists an α > 0 such that (1) λx 0 ∈ G − a         whenever    λ ∈ K   with   | λ | ≤ α . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1287.tif"/>

Now for any β ∈ K with |β — f(a)| ≤ α, we have (β — f(a))x₀ ∈ G — a (by(1)),hence β − f ( a ) = f ( ( β − f ( a ) ) x 0 ) ∈ f ( G ) − f ( a ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1288.tif"/>

which implies β ∈ f(G); thus f(a) is an interior point of f(G).

The following two separation theorems, that have become standard tools in functional analysis, are actually geometric forms of the Hahn—Banach extension theorem.

8.2 Theorem

(The first separation theorem). Let X be a TVS and A, B two non-empty convex sets in X. If Int   A   ≠ ϕ      and _      B ∩ Int   A = ϕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1289.tif"/> , then there exists a real 0 ≠ f ∈ X^' which separates A and B in the sense that (1) sup   f ( A ) [ = sup a ∈ A   f ( a ) ] ≤ inf   f ( B )   [ = inf b ∈ B   f ( b ) ] . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1290.tif"/>

Furthermore, if in addition, A and B are disjoint open sets, then A and B 133 are strictly separated by f in the sense that there is an λ ∈ ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1291.tif"/> such that f ( a ) < λ < f ( b ) ( or   f ( b ) < λ < f ( a ) )           ( for   all _   a ∈ A   and _   b ∈ B ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1292.tif"/>

Proof.

By (7.6 (b)), Int A is convex and open, hence —B + Int A is a nonempty open convex set which does not contain 0 [since B ⋂ Int A = ϕ]. Let e ∈ —B + Int A. Then —e —B + Int A is an open, convex 0—neighbourhood, hence its gauge, denoted by p, is a continuous sublinear functional on X such that (2) − e − B + Int   A = { x ∈ X : p ( x ) < 1 } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1293.tif"/> { λe : λ ∈ ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1294.tif"/> } is a real vector subspace of X, and the functional g, defined by g ( λe ) = − λ         ( for   all   λ ∈ ℝ ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1295.tif"/> ,

is a real linear functional on { λe : λ ∈ ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1296.tif"/> } such that (3) g ( λe ) ≤ p ( λe )         ( for   all   λ ∈ ℝ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1297.tif"/>

[Indeed, we first notice from (2) that p(—e) ≥ 1 since —e ∉ —e — B + Int A (on account of 0 ∉ —B + Int A); now for any λ ∈ ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1298.tif"/> , if λ > 0 then λp ( − e ) = p ( − λe ) ≥ λ = − g ( λe ) = g ( − λe ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1299.tif"/>

if λ < 0 then —λ > 0 and g ( λe ) = − λ ≤ − λp ( − e ) = p ( λe ) . ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1300.tif"/>

It then follows from Hahn—Banach′s extension theorem that there is a real 0 ≠ f ∈ X* such that f ( λe ) = g ( λe )         ( for   all   λ ∈ ℝ )     and     f ( x ) ≤ p ( x )         ( x ∈ X ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1301.tif"/>

134hence f is continuous on X [by the continuity of p]. For any b ∈ B and a ∈ Int A, we have f ( a − b ) = f ( − e + a − b ) + f ( e ) ≤ p ( − e + a − b ) + g ( e ) < ≠ 1 + g ( e ) = 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1302.tif"/>

thus (1) holds by the continuity of f.

Finally, we assume that A and B are disjoint open convex subsets of X. Then (8.1) shows that f(A) and f(B) are non-empty open convex subsets of ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1303.tif"/> with f(A) ⋂ f(B) = ϕ, and surely disjoint open intervals in ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1304.tif"/> ; therefore there is an λ ∈ ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1305.tif"/> such that f ( a ) < λ < f ( b )     ( or   f ( b ) < λ < f ( a ) )           ( for   all   a ∈ A   and   b ∈ B ) ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1306.tif"/>

namely, f separates strictly A and B.

From the first separation theorem, we see that for any two disjoint convex sets A and B, if they can be separated by some convex 0—neighbourhood V, that is (A + V) ⋂ (B + V) = ϕ, then we can apply the first separation theorem to get a non—zero f ∈ X^' strictly separating A and B. Therefore, it is important for us to find some sufficient condition ensuring that this can be done; (7.c) (iii) has already given such conditions.

8.3 Theorem

(The strong (or second) separation theorem). Let X be a LCS, and let K, B be two non-empty disjoint convex subsets of X. If K is compact and B is closed, then there exists a real 0 ≠ f ∈ X' and λ ∈ ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1307.tif"/> such that f strongly separates K and B in the sense that (8.3.1) max   f ( K ) < λ ≤ inf   f ( B )           ( or sup   f ( B ) ≤ λ < min   f ( K ) ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1308.tif"/>

Proof.

It is known [see(7.c)(iii) and (7.4)] that there is an open convex 0—neighbourhood V in X such that (K + V) ⋂ (B + V) = ϕ. As K + V and B + V are open and convex, it follows from (8.2) that there is a real 0 ≠ f ∈ X' and λ ∈ ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1309.tif"/> such that either   f ( K + V ) < λ < f ( B + V )     or     f ( B + V ) < λ < f ( K + V ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1310.tif"/>

135hence we have either f ( K ) < λ < f ( B + V )     or     f ( B + V ) < λ < f ( K ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1311.tif"/>

It then follows from the compactness of K that (8.3.1) holds.

Separation theorems have a lot of important applications, we mention some as follows:

8.4 Corollary

Let X be a LCS. For any 0 ≠ x₀ ∈ X, there is a 0 ≠ f ∈ X' such that f(x₀) ≠ 0.

Proof.

Follows immediately from (8.3) since { 0 } and { x₀ } are non-empty, disjoint compact convex sets.

Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1312.tif"/> be a LCS with the topological dual X'. The preceding result shows that { 0 } ≠ X' ⊂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1313.tif"/> X*, hence one can define the weak topology σ(X,X') on X (see (7.15) (b)). It is clear that σ(X,X') < P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1314.tif"/> hence every σ(X,X')—closed subset of X must be P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1315.tif"/> -closed. The converse is true for convex sets as shown by the following:

8.5 Corollary

Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1316.tif"/> be a LCS. Any. P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1317.tif"/> -closed convex subset B of X is σ(X,X')—closed; consequently, the P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1318.tif"/> -closure and the σ(X,X')—closure of any convex subset of X are identical.

Proof.

For any x₀ ∉ B, the strong separation theorem ensures that there exists a real 0 ≠ f ∈ X' and λ ∈ ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1319.tif"/> such that f ( x 0 ) < λ ≤ inf   f ( B ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1320.tif"/>

so that x₀ is an interior point of X\B for σ(X,X'),thus B is σ(X,X')—closed.

8.a <target id="page_136" target-type="page">136</target>Consequences of separation theorems :

(i) Let E be a normed space and B ⊂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1321.tif"/> E. If ‖ f ‖ = sup b ∈ B | f ( b ) |       ( for   all   f ∈ E ′ ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1322.tif"/>

then U E  =  ΓB ¯ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1323.tif"/>

(ii) Let X be a LCS and M a closed vector subspace of X. x₀ ∉ M if and only if there exists a 0 ≠ f ∈ X' such that f ( x 0 ) ≠ 0     and     M ⊂ f − 1 ( 0 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1324.tif"/>

Moreover, we have M = ( M ⊥ ) T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1325.tif"/> , where M ⊥ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1326.tif"/> is the annihilator of M, defined by M ⊥ = { f ∈ X ′ : M ⊂ f − 1 ( 0 ) } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1327.tif"/>

and ( M ⊥ ) T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1328.tif"/> (in X) is the annihilator of M ⊥ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1329.tif"/>

As another important application of separation theorems, we are going to verify Krein—Milman′s theorem. To do this, we need the following terminology:

Let X be a vector space and ϕ ≠ K ⊂ X . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1330.tif"/> A point a ∈ K is called an extreme point of K if the following holds: If x, y ∈ K are such that a = λx + ( 1 − λ ) y         for   some     λ ∈ ( 0 , 1 ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1331.tif"/>

then a = x = y. Denoted by ∂K the set consisting of all extreme points of K. More general, a non-empty subset B of K is called an extremal subset of K if the following holds: If x, y ∈ K are such that λx + ( 1 − λ ) y ∈ B           ( for   some   λ ∈ ( 0 , 1 ) ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1332.tif"/>

then x,y ∈ B.

8.b A characterization of extreme points:

137Let X be a vector space, let K be a non-empty convex subset of X and u_o ∈ K. Then the following statements are equivalent:

u_o is an extreme point of K.

If x,y ∈ K are such that u₀ = 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1333.tif"/> (x+y), then u_o = x = y.

Let x,y ∈ K be such that x ≠ y, let λ∈[0,1] and u_o=λx+(1—λ)y. Then we have either λ = 0 or λ = 1.

K\{X_o} is convex.

Although the notion of extreme points is purely algebraic, we have the following interesting result, which is useful to give examples of extreme points of closed unit balls in some concrete Banach spaces.

8.6 Proposition

Let X be a TVS and K a convex subset of X. Then (IntK) ⋂ (∂K) = ϕ [i.e., any interior point of K (when it exists) is not an extreme point of K.]

Proof.

If IntK = ϕ, the result is trivial. Suppose now that IntK ≠ ϕ and that x ∈ IntK. Then there exists a 0—neighbourhood V in X such that x + V ⊂ K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1334.tif"/> . As the map μ→μx : ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1335.tif"/> → X is continuous at μ= 1, for this x—neighbourhood x + V, there is an r > 0 such that μ x ∈ x + V     whenever     ​ | μ − 1 | ≤ r ( μ ∈ ℝ ) ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1336.tif"/>

in particular, we have ( 1 + r ) x ∈ x + V ⊂ K         and       ( 1 − r ) x ∈ x + V ⊂ K . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1337.tif"/>

Consequently, we have x = 1 2 ( 1 + r ) x + 1 2 ( 1 − r ) x , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1338.tif"/>

which implies that x is not an extreme point of K.

138As a consequence of (8.6), we see that any open convex subset of a TVS does not have any extreme point.

8.7 Examples

(a) In the Banach space ℓ n ∞ ( ℝ ) = ( ℝ n , ‖ ⋅ ‖ ∞ ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1339.tif"/> , let U = { [ ζ i ] ∈ ℝ n : ‖ [ ζ i ] ‖ ∞ = max 1 ≤ i ≤ n | ζ i | ≤ 1 } ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1340.tif"/> one has ∂ U = { [ ζ i ] ∈ U : | ζ i | = 1       for   i = 1 , .. , n } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1341.tif"/>

Proof.

By (8.6), if [ζ_i] ∈ ∂U then ‖ [ ζ i ] ‖ ∞ = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1342.tif"/> , moreover we claim that |ζ_i| = 1 (i=l,⋯,n). Suppose not, there is some k ∈ {l,⋯,n} such that |ζ_k| < 1. Then one can choose r > 0 such that |ζ_k| +r < 1. Now consider the following two vectors y = ( ζ 1 , .. , ζ k − 1 , ζ k + r , ζ k + 1 , .. ζ n )       z = ( ζ 1 , .. , ζ k − 1 , ζ k − r , ζ k + 1 , .. ζ n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1343.tif"/>

Then y, z ∈ U are such that y ≠ z       and       [ ζ i ] = 1 2 ( y + z ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1344.tif"/>

which contradicts the fact that [ζ_i] ∈ ∂U.

Conversely, let [ζ_i] ∈ U be such that |ζ_i| =1 (i = l,⋯,n) and let [λ_i], [μ_i] ∈ U be such that [ ζ i ] = 1 2 [ λ i ] + 1 2 [ μ i ] . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1345.tif"/>

Then for any i ∈ {l,⋯,n}, 1 2 λ i + 1 2 μ i = 1         or       − 1. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1346.tif"/>

As 1,−1 are extreme points of [−1,1], it follows that λ i = μ i = 1         or       − 1 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1347.tif"/>

139and hence that [ζ_i] = [λ_i] = [μ_i]; in other words, [ζ_i] ∈ ∂U.

(b) In the Banach space l ¹ we have ∂ U l 1 = { λe ( n ) : λ ∈ ℂ , | λ | = 1    and    n = 1 ⋯ } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1348.tif"/>

Proof.

Suppose that |λ| =1 and λe⁽ⁿ⁾ = 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1349.tif"/> (y+z), where y=[y_i], z=[z_i] ∈ U _{l ¹} . Then (1) λ = 1 2 y n + 1 2 z n      and      0 = 1 2 y i + 1 2 z i ( i ≠ n ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1350.tif"/>

As λ (with |λ| = 1) is an extreme point of U_K, it follows that λ = y_n = z_n, and hence from 1 = | λ | ≤ ∑ k = 1 ∞ | y k | ≤ 1    and    1 = | λ | ≤ ∑ k = 1 ∞ | z k | ≤ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1351.tif"/>

that all y_i = z_i = 0 (for i ≠ n); consequently, λe ( n ) = y = z ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1352.tif"/>

thus λe⁽ⁿ⁾ ∈ ∂U _{l ¹} .

Conversely, if [ζ_i] ∈ ∂U _{l ¹} , then (8.6) shows that ||[ζ_i]||₁ = 1; moreover, we claim that there is some n such that ζ i = 0       for   all   i ≠ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1353.tif"/> (hence [ζ_i] = ζ_ne⁽ⁿ⁾ with |ζ_n| = 1 (since ||[ζ_i]||₁ = 1)).

In fact, suppose, on the contrary, that there are j and k with j < k such that (ζ_j ≠ 0 and ζ_k ≠ 0. Then 0< ∑ i = 1 j | ζ i | = μ < 1       and       ∑ i > j ∞ | ζ i | = 1 − μ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1354.tif"/>

140(since ‖ [ ζ i ] ‖ 1 = ∑ i = 1 ∞ | ζ i | = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1355.tif"/> ). Let y = 1 μ ( ζ 1 ⋯ , ζ j , 0 , 0 ⋯ )       and       z = 1 1 − μ ( 0 , ..0 , ζ j+1 , ⋯ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1356.tif"/>

Then y, z ∈ U _{l ¹} are such that y ≠ z       a n d       [ ζ i ] = μ y + ( 1 − μ ) z , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1357.tif"/>

where μ ∈ (0,1); this contradicts the assumption that [ζ_i] ∈ ∂U _{l ¹} .

(c) In the Banach space c_o, we have ∂ U c o = ϕ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1358.tif"/>

Proof.

It suffices to show that any 0 ≠ [ζ_i] ∈ U_co is not an extreme point of U_co

Indeed, since lim j ζ j = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1359.tif"/> , it follows from ||[ζ_i]||_∞ ≠ 0 that there is some k ∈ N such that | ζ k | < ‖ [ ζ i ] ‖ ∞ ≤ 1 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1360.tif"/>

and hence that one can choose r > 0 such that 0 < | ζ k | + r < 1. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1361.tif"/>

Now consider the vectors y = [y_i] and z = [z_i] with y i = { ζ i                         if   i ≠ k ζ k + r         if   i = k           z i = { ζ i                         if   i ≠ k ζ k − r         if   i = k . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1362.tif"/>

then y,z ∈ U_co are such that y ≠ z       and       [ ζ i ] = 1 2 y + 1 2 z , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1363.tif"/>

hence [ζ_i] is not an extreme point of U_co .

141(d) In the Banach space L¹( ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1364.tif"/> ), one has ∂ U L 1 ( R ) = ϕ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1365.tif"/>

Proof.

By (8.6), it suffices to consider any u ∈ U_{L¹( ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1366.tif"/> )} with ||u||₁ = 1 not extreme point of U_{L¹( ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1367.tif"/> )}. Indeed, one can choose r ∈ ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1368.tif"/> such that ∫ − ∞ r | u ( t ) | dm ( t ) = 1 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1369.tif"/>

Now let y = 2u ℵ ( − ∞ , r )         and         z     =2u ℵ [ r,+ ∞ ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1370.tif"/>

where ℵ ( − ∞ , r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1371.tif"/> and ℵ [ r,+ ∞ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1372.tif"/> are respectively the characteristic functions of (—∞,r) and [r,+∞).

Then ||y||₁ = ||z||₁ are such that y ≠ z       and       u = 1 2 y + 1 2 z, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1373.tif"/>

hence u is not an extreme point of U_{L¹( ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1374.tif"/> )}.

8.c

Extreme points of closed unit balls in the concrete B–spaces:

(i) For 1 < p < ∞, one has ∂ U L p ( ℝ ) = { f ∈ L p ( ℝ ) : ‖ f ‖ p = 1 }     and ∂ U ℓ p = { | ς n | ∈ ℓ p ( ℝ ) : ‖ | ς n | ‖ p = 1 } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1375.tif"/>

(ii) ∂ U L ∞ [ 0 , 1 ] = { f ∈ L ∞ [ 0 , 1 ]   :   | f ( t ) | = 1  almost all t ∈ [ 0 , 1 ] } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1376.tif"/> . In l ^∞, let A = { e ( n ) n : n ≥ 1 } ∪ { 0 }  and K = coA ¯ ..  Then  ∂ K = A . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1377.tif"/>

(iii) Let Ω be a compact Hausdorff space, and let M(Ω) be the Banach space of all complex regular Borel measures [see (3.9)(f)]. Then ∂ U M ( Ω ) = { λ δ t     :     λ ∈ ℂ , | λ | = 1   and   t ∈ Ω } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1378.tif"/>

142where δ_t ∈ M(Ω) is the Dirac measure on Ω, i.e., δ t ( B ) = { 1         if   t ∈   B 0         if   t ∉   B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1379.tif"/>

for any Borel set B  ⊂   Ω https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1380.tif"/> .

(iv) ∂ U C ℝ ( [ 0 , 1 ] ) = { x ∈   C ℝ [ 0 , 1 ]       :       x ( t ) ≡ 1     or     − 1     ( t ∈ [ 0 , 1 ] ) } ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1381.tif"/> and more generally one can show that ∂ U C ( Ω ) = { x ∈   C ( Ω ) ; | x ( t ) | = 1 ( ∀   t   ∈   Ω ) } ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1382.tif"/>

where Ω is a compact Hausdorff space.

8.8 Lemma

Let X be a LCS and K a non-empty compact convex subset of X. For anv real 0 ≠ f ∈ X', the set, defined bv K ( f ) = { x ∈ K   :f ( x )   =   int   f ( K ) } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1383.tif"/>

is a non-empty, convex, extremal subset of K (called a face in K).

Proof.

The convexity of K and the linearity of f imply that K_f is convex, which is non-empty (by the compactness of K). On the other hand, the definition of infimum and the linearity of f imply that K_f is extremal.

8.9 Theorem (Krein—Milman)

Let X be a LCS and K a non-empty, compact convex subset of X. Then

(a) C₀ ⋂ ∂K ≠ ϕ for anv non-empty closed, convex, extremal subset C₀ of K (called a closed face in K).

(b) K = co ¯ ( ∂ K ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1384.tif"/> . Consequently, anv non-empty compact convex set in a LCS always has extreme points.

Proof.

(a) Let E = { C ⊂ K : C closed face in K with C ⊆ C 0 } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1385.tif"/> . Then E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1386.tif"/> is an inductive ordered set under downward inclusion. [If M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1387.tif"/> is a chain in E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1388.tif"/> and if C α i = M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1389.tif"/> 143(i = 1,2,⋯,n), then ∩ i = 1 n C α i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1390.tif"/> is one of C α 1 , ⋯ , C α n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1391.tif"/> , that is M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1392.tif"/> has the finite intersetion property, thus ∩ M ≠ ϕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1393.tif"/> by the compactness of K; consequently ∩ M ∈ E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1394.tif"/> is such that ∩ M ≤ C α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1395.tif"/> (for all C α ∈ M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1396.tif"/> ).] By Zorn′s lemma, E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1397.tif"/> contains a minimal element C. (Remember that C is a closed face in K with C ⊆ C 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1398.tif"/> ). Now we show that C = { e }   for   some   e ∈ K, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1399.tif"/>

thus e ∈ ∂K and e ∈ ∂K ⋂ C₀ (on account of e ∈ C ∩ ∂ K ⊂ C 0 ∩ ∂ K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1400.tif"/> ).

In fact, if C contains elements x and y with x ≠ y, then (8.3) shows that there is a real 0 ≠ f ∈ X' such that f(x) ≠ f(y). By (8.6), the set C f = { z   ∈   C   :   f ( z ) = inf ( C ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1401.tif"/>

is a closed face in C, (hence in K) with C f ⊂ C 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1402.tif"/> . As f(x) ≠ f(y) it follows that C f ⊂ ≠ C https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1403.tif"/> , which contradicts the minimality of C.

(b) Let D = co ¯   ∂ K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1404.tif"/> . Then D ⊂ K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1405.tif"/> .If D ≠ K, one can take x₀ ∈ K\D. By (8.3), there is a real 0 ≠ h ∈ X^' such that (1) h ( x 0 ) < min   h ( D ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1406.tif"/>

On the other hand, (8.8) shows that the set, defined by K h = {   z ∈ K : h ( z ) = inf   h ( K )   } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1407.tif"/>

is a closed face in K, thus there is an e ∈ (∂K) ⋂ K^h [by part (a)]; consequently, (2) h ( e ) = inf   h ( K ) ≤ h ( x 0 ) < min   h ( D ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1408.tif"/>

[by(l)]. Notice that e ∈ ∂ K ⊂ D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1409.tif"/> , so that min   h ( D ) ≤ h ( e ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1410.tif"/>

which contradicts (2).

<target id="page_144" target-type="page">144</target>Remark

The condition that X be a locally convex space cannot be dropped from the above theorem; in fact, Roberts[1977] (see Kalton/Peck [1980]) gives an example of a compact convex subset of L p [ 0 , 1 ] , 0 < p < 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1411.tif"/> , with no extreme points.

Example (e)

Even in a finite—dimensional Banach space, the set of extreme points of a compact, convex set K need not be closed. For instance, in ℝ 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1412.tif"/> , let D = { ( x,y,z ) ∈ ℝ 3 : z = 0 ,   ( x − 1 ) 2 + y 2 = 1 } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1413.tif"/>

and K = co ( D ∪ { ( 0 , 0 , 1 ) , ( 0 , 0 , − 1 ) } ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1414.tif"/>

then K is compact convex and ( 0 , 0 , 0 ) ∈ ∂ K ¯ \ ( ∂ K ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1415.tif"/> , hence ∂K is not closed.

As a consequence of Krein — Milman's theorem, we obtain:

8.d

Let X, Y be LCS, let T ∈ L(X,Y) and K a non-empty compact convex subset of X. Then ∂ ( T ( K ) ) ⊂ T ( ∂ K ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1416.tif"/> .

8.e

A partial converse of Krein — Milman's theorem is the following:

(8.e) Let X be a LCS and ϕ ≠ A ⊂ X.

(i) (Milman) If c o ¯  A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1417.tif"/> is compact, then ∂ ( c o ¯  A ) ⊂ A ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1418.tif"/> .

(ii) Let K ⊂ X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1419.tif"/> be compact convex with A ⊂ K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1420.tif"/> . Then K = c o ¯  A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1421.tif"/> if and only if ∂ K ⊂ A ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1422.tif"/> consequently, ∂ K ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1423.tif"/> is the minimal closed subset of I\ whose closed convex hull is K.

8.f Extrema at extreme points:

Let X be a real LCS and K a non-empty compact convex subset of X.

For any 0 ≠ f ∈ X^, there exists an e ∈ ∂K such that f ( e ) = max   f ( K ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1424.tif"/>

145Moreover, ∂ K ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1425.tif"/> is the smallest closed subset of K on which every 0 ≠ f ∈ X^, attains its maximum over K.

(Bauer) For any upper semicontinuous convex function g : K → ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1426.tif"/> , there is a u ∈ ∂K such that g ( u ) = max   g ( K ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1427.tif"/>

For any lower semicontinuous concave function h : K → ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1428.tif"/> , there is a w ∈ ∂K such that h ( w ) = min  h ( K ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1429.tif"/>