By Robert E. Megginson

Getting ready scholars for additional examine of either the classical works and present study, this is often an available textual content for college students who've had a direction in genuine and complicated research and comprehend the fundamental homes of L p areas. it truly is sprinkled liberally with examples, historic notes, citations, and unique assets, and over 450 workouts offer perform within the use of the consequences constructed within the textual content via supplementary examples and counterexamples.

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**Example text**

The collection of all sequences (a:;) of scalars for which E~ 1 la:; IP is finite is a vector space with the vector space operations of the preceding example. Let the norm ll·llp be defined on this vector space by the formula The resulting normed space is called ip (pronounced "little ell p"). 9 Example: £;, 1 :5 p :5 oo. Let p be such that 1 5 p :5 oo and let n be a positive integer. Define a norm on the vector space lFn by letting ll(a:l,···•O:n)llp= { ( I)a:;IP) l/p i=l max{la:1l, ... ,la:nl} if 1 :5 p < oo; if p = oo.

14 Example: Lp('ll') and Hp, 1 ~ p ~ oo. Fix p such that 1 ~ p ~ oo. Let 1l' be the unit circle { z: z E C, lzl = 1} in the complex plane, and let X be the set of all complex-valued functions f on 1l' with the property that if g: [-1r, 1r) ~ C is defined by letting g(t) = f(eit), then g E Lp[-1r, 1r). As would be expected, two functions fi and h in X are considered to be the same if JI(eit) = h(eit) for almost all tin [-1r, 1r). Since Lp[-1r, 1r) is a Banach space, it is clear that X is also a Banach space with the obvious vector space operations and the norm ll·llp given by letting 11/llp = { ( 2~ i: ll9lloo lf(eit)JP dt r/P (where g is as above) if 1 ~ p < OOj if p = oo.

3 Definition. Let X andY be normed spaces. For each Tin B(X, Y), the norm or operator norm IITII of T is the nonnegative real number sup{ 1\Txl\: x E Bx }. The operator norm on B(X, Y) is the map T ~--+ IITII. 4 Example. Let X and Y be normed spaces. The zero operator from X into Y is the zero element of the vector space B(X, Y), that is, the operator that maps each x in X to the zero element of Y. This operator clearly has norm 0. The identity operator on X is the member I of B(X) defined by the formula Ix = x.