Normed Linear Spaces
Each of the function spaces mentioned in the introduction of th preceding chapter has (with one exception, A(D)) a norm ∥ ∥ [Definition I, 3, 1] which defines the topology of major interest in the space; a neighborhood basis of a point x is the family of
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§ 1. Elementary Definitions and Properties Each of the function spaces mentioned in the introduction of the preceding chapter has (with one exception, A(D)) a norm 11 11 [Definition I, 3, 1] which defines the topology of major interest in the space; a neighborhood basis of a point x is the family of sets {y: 11 x - y 11 ~ I>} where 1»0. Hereafter we shall use N for a normed space, that is, a linear space in which a norm is already assigned. If the normed space is completel under the metric 11 x - y 11, then the space will be called a Banach space, and will generally be denoted by B. U will generally stand for the unit ball, {x: 11 x 11 ~ 1 }, unless otherwise noted; the unit sphere is the set {x: 11 x 11 = 1}. The properties we have discussed in linear topological spaces sometimes have simpler character in normed spaces. (1) A set E in a normed space is bounded if and only if it lies in some ball. (2) Call a linear operator T from one normed space N into another N' bounded if it is bounded on the unit ball U in N; define 11 T 11 , the bound or norm of T, to be sup{IITxll: Ilxll~1}. Then: (a) IITxl1 ~ 11 Tll-llxll if xEN. (b) IITII =sup{11 Txll: Ilxll =1} =sup{11 Txll: Ilxll < 1} =sup{IITxll/llxll:x=\=O}. (c) T is bounded if and only if it is continuous, and if and only if it is uniformly continuous. (d) If T' is a bounded linear operator from N' into N", then 11 T' T 11 ~ 11 T' 11-11 TII· (e) If T is a bounded linear operator from N into B', and if B is the completion of N, then T has a unique continuous linear extension T' from B into B', and 11 T'II = 11 TII· (3) Let ~(N, N') be the space of all continuous linear operators from N into N'. (a) ~(N,N') is a normed linear space. (b) ~(N,N') is a Banach space (that is, complete) if and only if N' iso (c) Hence N* is a Banach space. (d) This norm determines the '!3" topology in N*. 1
That is, every Cauchy sequence in N has a limit in N.
M. M. Day, Normed Linear Spaces © Springer-Verlag Berlin Heidelberg 1973
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Chapter 11. Normed Linear Spaces
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(e) Every bounded set E in 52(N,N') is a uniformly equi-continuous set of functions on N into N'; that is, given ß > there is () > such that for all Tin E and all x,y in N IITx-Tyll 0)-t0. (5) To prove from these axioms that M is an LTS go es in several steps. (a) (BI) implies continuity of addition. (b) (BI) and (B 2 ) imply that multiplication is continuous in the first variable; (c) (BI) and (B 3 ) give continuity in the second variable. (d) A category argument on the sequence of real-valued functions d(axn>O), lai ~ 1, gives continuity of multiplication at (0,0). (See § 3 for category proofs.) (e) (BI) and (d) give continuity of multiplication, the only missing condition. (1) Replacing d by a new metric d', defined by d'(x,O) = sup{d(ax,O): lal~1} and d' (x, y) = d' (x - y, 0) gives a metric which is topologically and uniformly equivalent to the old but has the extra property that d' (a x, 0) is a nondecreasing function of Ia!. (6) Any invariant metric in an LTS L which yields the original topology of L also yiel
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