Fabian - Banach space theory (The basis for linear and nonlinear)

[...]... 1, 0, ), and 1 is at the ith position These vectors are called the canonical unit vectors Proposition 1.16 (i) For p ∈ [1, ∞], the space p is a Banach space 8 1 Basic Concepts in Banach Spaces (ii) The spaces c and c0 are closed subspaces of spaces (iii) The space c00 is not complete ∞ and thus they are Banach In the proof we will use the following lemma Lemma 1.17 Let X be a normed space If a sequence... 10.1007/97 8-1 -4 41 9-7 51 5-7 _1, C Springer Science+Business Media, LLC 2011 1 2 1 Basic Concepts in Banach Spaces All topological and uniform notions in normed spaces refer to the canonical metric given by the norm, unless stated otherwise In situations when more than one normed space is considered, we will sometimes use · X to denote the norm of X Definition 1.2 A Banach space is a normed linear space (X,... X , then span(M) stands for the linear hull—or span—of M, that is, the intersection of all linear subspaces of X containing M Equivalently, span(M) is the smallest (in the sense of inclusion) linear subspace of X containing M, or the set of all finite linear combinations of elements in M Similarly, span(M) stands for the closed linear hull of M, i.e., the smallest closed linear subspace of X containing... Spaces On [0, 1]\ 13 Z k,l the functions { f n } form a uniformly Cauchy and therefore unik,l formly convergent sequence From this and from the completeness of the supremum norm, we obtain that L ∞ is a Banach space The space L ∞ is infinite-dimensional; to see this follow the same argument as in the L p case Given a measure space (Ω, μ) and p ∈ [1, ∞], the space L p (Ω, μ) (also denoted L p (μ)) can be... 2.43 Proposition 1.27 Let X, Y be normed linear spaces If Y is a Banach space then B(X, Y ) is also a Banach space Proof: The proof of completeness is similar to that for the space C[0, 1] In particular, if {Tn } is a sequence of linear mappings from X into Y and Tn (x) → T (x) for all x ∈ X , then T must be linear Operators in B(X, K) are called continuous linear functionals The operator norm introduced... shown above that, for every n ≥ n 0 , f ∞ ≤ ε This proves that f n − f ∞ → 0, so C[0, 1] is complete fn − Analogously, the space C(K ) of continuous scalar functions on a compact space K , endowed with the supremum norm, is a Banach space 4 1 Basic Concepts in Banach Spaces We note that C[0, 1] is an infinite-dimensional Banach space To see this, it is enough to produce, for any n ∈ N, a linearly independent... Quotients, Finite-Dimensional Spaces We now begin an investigation of linear mappings between normed spaces Recall that a mapping T from a vector space X over the field K into another vector space Y over K is called linear if T (α1 x 1 + α2 x2 ) = α1 T (x1 ) + α2 T (x 2 ) for every α1 , α2 ∈ K and x1 , x2 ∈ X The vector space of all linear mapping from X into Y will be denoted L(X, Y ) A linear functional... misunderstanding can arise, by a “subspace” of a vector space we will mean a linear subspace and, in case of normed spaces, a closed linear subspace Definition 1.3 Let E be a vector space Given x, y ∈ E, the set [x, y] := {λx + (1 − λ)y : 0 ≤ λ ≤ 1} is called the closed segment defined by x and y If x = y, the set (x, y) := {λx + (1 − λ)y : 0 < λ < 1} is called the open segment defined by x and y A subset... the subspaces {(x, 0) : x ∈ X } and {(0, y) : y ∈ Y } of X ⊕ Y , respectively Definition 1.33 Let (X, · X ) and (Y, · Y ) be normed spaces The algebraic direct sum X ⊕ Y of X and Y becomes a normed space, called the topological direct sum of X and Y and still denoted X ⊕ Y , when it is endowed with the norm (x, y) := x X + y Y The spaces X and Y are isometric to the subspaces {(x, 0) : x ∈ X } and {(0,... coordinatewise) For subsets A, B of a vector space X and a scalar α we also write A + B := {a + b : a ∈ A, b ∈ B} and α A := {αa : a ∈ A} A set M ⊂ X is called symmetric if (−1)M ⊂ M, and balanced if α M ⊂ M for all α ∈ K, |α| ≤ 1 Let Y be a subspace of a normed space (X, · ) By (Y, · ) we denote Y endowed with the restriction of · to Y if there is no risk of misunderstanding Fact 1.5 Let Y be a subspace of a Banach . Mathematics and Statistics Dalhousie University Halifax, Nova Scotia B3H 3J5 cbs-editors@cms.math.ca ISSN 161 3-5 237 ISBN 97 8-1 -4 41 9-7 51 4-0 e-ISBN 97 8-1 -4 41 9-7 51 5-7 DOI 10.1007/97 8-1 -4 41 9-7 51 5-7 Springer. in Banach space theory and introduce the classical Banach spaces, in particular sequence and function spaces. v vi Preface In Chapter 2 we discuss two fundamental principles of Banach space theory, namely. to http://www.springer.com/series/4318 Marián Fabian · Petr Habala · Petr Hájek · Vicente Montesinos · Václav Zizler Banach Space Theory The Basis for Linear and Nonlinear Analysis 123 Marián Fabian Mathematical Institute