Functions whose values are sets in a vector space
James Frederick Leetch · OhioLink ETD Center (Ohio Library and Information Network) · 1961
Definition 2.2 B is a normed linear space (over K): i) B is a linear space (over K), ii) There exists a real valued function Hx'i , x 6 B, called the norm of x, satisfying: i l 011 = 0 IIx U > 0 for x e B, x 4 0 I I x + y I I ^ I I xn t nyil for x, y 0, there exists N such that m, n > N implies I i -Xjj i \ ^ , x_ € B, has a limit in B: There exists x e B such that lim x =x, i.e., for every t >0, there exists N such that n > N implies I I xn -X H -c £ .) 1* We will now be concerned with the family of all non-empty, bounded subsets of a normed linear or Banach space B. We designate this family by B and the members by X, Y, Z, ... (X is bounded: There is a real number R such that uxu g R, for all x G X) The complement of X relative to B is denoted X 1. Open sets in B result from the natural topology given by the norm.The closure of K B is denoted by X. Definition 2.I t For X, Y B and a c K we define:I® 1 in general.The conditions for equality and the other results stated in theorem 2.1 follow directly from the definitions.Theorem 2.1 For X, Y, Z < £ B and a, b G K : X i Y = Y -+ X, X -r (I 4 Z) (X 1 Y) • Z .a(X -f Y) -aX -f aX , I I aX -+ -bY I I -i ai I i X H -h I b I UY 1 1 , i!aX -bill ^ |iai IIX i l -Ibt ill H1 , U X 1 1 -UXi( aX ~ a X , X -X = 0 if and only if X is a set consisting of a single element of B, X = X, X ' Y -X + Y .Definition 2.5 C [xl , X ^ B , the convex hull of X : | " 2^ * ■ * * ® j .~ 1 ^ X , n = 1, 2, 3) •••! •