Functional Analysis (6th Edition)

Author: K.Yosida
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Publish Date: 1999-06-01
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Fragment: 0. Preliminaries It is the purpose of this chapter to explain certain notions and theorems used throughout the present book. These are related to Set Theory, Topological Spaces, Measure Spaces and Linear Spaces.
1. Set Theory
Sets. x ∈ X means that x is a member or element of the set X; x ? X means that x is not a member of the set X. We denote the set consisting of all x possessing the property P by {x; P}. Thus {y; y = x} is the set {x} consisting of a single element x. The void set is the set with no members, and will be denoted by ?. If every element of a set X is also an element of a set Y, then X is said to be a subset of Y and this fact will be denoted by X ? Y, or Y ? X. If X is a set whose elements are sets X, then the set of all x such that x ∈ X for some X ∈ X is called the union of sets X, and this union will be denoted by ∪X∈X X. The intersection of sets X is the set of all x that are elements of every X ∈ X; this intersection will be denoted by ∩X∈X X. Two sets are disjoint if their intersection is void. A family of sets is disjoint if every pair of distinct sets in the family is disjoint. If a sequence {Xn}n=1,2,... of sets is a disjoint family, then the union X may be written in the form of a sum.
Mappings. The terms mapping, function, and transformation will be used synonymously. The symbol f: X → Y will mean that f is a single-valued function whose domain is X and whose range is contained in Y; for every x ∈ X, the function f assigns a uniquely determined element f(x) = y ∈ Y. For two mappings f: X → Y and g: Y → Z, we can define their composite mapping g f: X → Z by (g f)(x) = g(f(x)). The symbol f(M) denotes the set {f(x); x ∈ M} and f(x)) is called the image of M under the mapping f. The symbol f?1(N) denotes the set {x; f(x) ∈ N} and f?1(N) is called the inverse image of N under the mapping f. It is clear that...
Yosida. Functional Analysis. If f: X → Y, and for each y ∈ f(X) there is only one x ∈ X with f(x) = y, then f is said to have an inverse (mapping) or to be one-to-one. The inverse mapping then has the domain f(X) and range X; it is defined by the equation x = f?1(y) = f?1({y}). The domain and the range of a mapping f will be denoted by D(f) and R(f), respectively. Thus, if f has an inverse, then... The function f is said to map X onto Y if f(X) = Y and into Y if f(X) ? Y. The function f is said to be an extension of the function g and g are said to be strictly on f if D(f) contains D(g), and f(x) = g(x) for all x ∈ D(g).
Zorn's Lemma. Definition. Let P be a set of elements a, b,.... Suppose there is a binary relation defined between certain pairs (a, b) of elements of P, expressed by a R b if a R a and a R b. Then P is said to be partially ordered (or semi-ordered) by the relation R.
Examples. If P is the set of all subsets of a given set X, then the set inclusion relation (A ? B) gives a partial ordering of P. The set of all complex numbers z = x + iy, where x, y ∈ u + iv,... is partially ordered by defining z? ≤ z? if...
Definition. Let P be a partially ordered set with elements a, b,... If a bound or the supremum of a and b exists, and write c = sup(a, b) or a ∨ b, then this element of P is unique if it exists. In a similar way, we define the greatest lower bound or the infimum of a and b, and denote it by inf(a, b) or a ∧ b. If a ∨ b and a ∧ b exist for every pair (a, b) in a partially ordered set P, P is called a lattice.
Example. The totality of subsets M of a fixed set B is a lattice by the partial ordering M? ? M?.