Closed Subsets

Let K cV he a. convex closed subset of a reflexive Banach space V, I he a duality mapping, and P be a projection operator of V onto K. We are in a position to give a definition of a penalty operator. An operator (5 V V is called a penalty operator connected with the set K if the following conditions are fulfilled. Firstly, / is a monotonous bounded semicontinuous operator. Secondly, a kernel of / coincides with K, i.e. 

Proof. First suppose is a complex torus T. Taking a basis of V, we may suppose we are in the above situation. (If some coordinates are 0, we replace P by a subspace.) We may also assume that Y is the unique closed orbit in the closure of x. The uniqueness follows from the existence of T -invariant polynomial which separates two disjoint T -invariant closed subsets (Theorem 3.3). 

Goodykoontz, J. T., Jr. (1981) Hyperspaces of arc-smooth continua. Houston Journal of Mathematics. 7(1) 33—41. (Discusess the hyperspace of closed subsets.)... 

Now we prove Assertion (5). Suppose Gc (x, z) is closed. Since Gc- (x, z) and V x 0 are mutually disjoint, closed subsets, there exists an invariant polynomial P = zPi(x) + ----h znPn(x) which satisfies... 

XVIH) Let f 35 — s be a morphism of finite type and let F be a coherent sheaf on X. There exists a partition of 5 into locally closed subsets S, such that, giving each Sj the reduced scheme structure, the restrictions of F to each XxsSt is flat over 5, for all i. 

If U = 5 the theorem is true. Let s assume U 5. Note that for every irreducible closed subset V of codimension one in 5, letting v,s be the generic points of V and of 5 respectively, we have ... 

As mentioned in the preface of this monograph closed subsets play an important role in scheme theory. Via the group correspondence, closed subsets generalize the notion of a subgroup. They also generalize some of the important properties of subgroups from group theory to scheme theory. 

The first section of this chapter is a collection of general observations on closed subsets most of which are straightforward generalizations of facts on subgroups. For instance, we show that the set of all double cosets of two closed subset of S is a partition of S cf. Lemma 2T.3. We also introduce transversals (as they have been introduced in [45]), and, at the end if this section, we show that closed subsets give rise to subschemes. 

In the third section of this chapter, we investigate the relationship between closed subsets of S and structure constants of S. 

In Section 2.4, we apply some of the previously obtained results on closed subsets in order to derive a sufficient condition for a closed subset to be maximal. 

In Section 2.5, we define the normalizer and the strong normalizer of closed subsets. In the last of the six sections of this chapter, we introduce conjugates of closed subsets. Conjugates are related to normalizers and strong normalizers and will play a role in Section 4.4 when we investigate Sylow subsets. 

There are a few elementary facts about closed subsets which we occasionally shall quote without reference. Let us look at these facts first. 

Note finally that, as each closed subset of S contains 1 as an element, intersections of closed subsets of S are closed. 

Lemma 2.1.1 LetT and U be closed subsets of S. ThenTU is closed if and only ifTU = UT. 

Lemma 2.1.2 LetT and U be closed subsets of S. Then we have 1 = TCiU if and only if, for each element s in TU, there exist uniquely determined elements t in T and u in U such that s tu. 

Theorem 2.1.5 For each closed subset T of S, we have the following. 

Corollary 2.1.7 LetT be a closed subset ofS, and assume thatTs has finite valency. Then Ts is closed. 

We shall now see that closed subsets of S give rise to new schemes, the so-called subschemes. 

Lemma 2.3.1 Let p and q be elements in S, let T be a closed subset of S, and assume that T has finite valency. Then the following hold. 

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