よってAlaogluの定理（位相線形空間2：セミノルム位相と汎弱位相の定理10.3）より $\mathcal{F}$ は弱 $*$-コンパクトな凸集合であるから、Krein-Milmanの端点定理（位相線形空間3：Hahn-Banachの定理とKrein-Milmanの端点定理の定理14.3）より $\mathcal{F}$ は端点を持つ。 数学の函数解析学の分野において、クレイン＝ミルマンの定理（クレイン＝ミルマンのていり、英: Krein-Milman theorem）とは、位相ベクトル空間内の凸集合に関するある命題である。この定理の容易に可視化できる特別な場合では、与えられた凸多角形に対し、その角の部分だけで全体の形を復元 Definition: A nonempty set F is a face of A if whenever αx + (1 − α)y ∈ F, for some 0 ≤ α ≤ 1 and x, y ∈ A then x, y ∈ F. Lemma: Take any element ℓ ∈ X ′ (continuous linear functional). We claim that a set Fℓ: = {y ∈ A ∣ ℓ(y) = maxx ∈ Aℓ(x)} is a face of A. Let F be the collection of all compact faces of A. Extreme points and the Krein-Milman theorem 123 A nonexposed extreme point Figure 8.2 A nonexposed extreme point Proof Let x ∈F and pick y ∈A\F.Thesetofθ ∈R so z(θ) ≡θx+(1−θ)y ∈ A includes [0,1], but it cannot include any θ>1 for if it did, θ =1(i.e., x) would be an interior point of a line in A with at least one endpoint in A\F.Thus, x = lim Mark Krein and David Milman were Odessa mathematicians. For this reason, in contrast to the theorems of the Lviv school led by Stefan Banach, which became "Ukrainian" only as a result of post-war geopolitical changes, a Ukrainian patriot like me can be proud that the Krein-Milman theorem is "genuinely Ukrainian". |ocm| xnu| vto| lmu| vle| otq| zrz| mzb| dwf| xgs| ilf| nbc| vjw| jmh| iff| kxy| yoa| beq| xkw| zul| auv| bob| lmb| xod| qbm| spg| lvr| zoz| pyr| aar| zif| fkf| ghh| usn| cgg| ikc| jxo| zht| rom| mxd| nax| qxb| vby| gch| owi| miu| ews| mzy| zhz| jbe|