12

W. C. CONNETT AND A. L. SCHWARTZ

CHAPTER I

THE INTERPOLATION OF "LOCAL" SPACES

1. Notations and Definitions

1.1 If B is a normed linear space and x € B, the norm of x is denoted

by B[x]. If A is a second normed linear space T:A - » B means T is a

bounded linear transformation from A to B (although continuity will often

be mentioned explicitly as well). If A c = B, and if B[x] MA[x] for some

positive constant M independent of the choice of x € A write

A B and B[x] A[x];

if both A B and B A, write

A « B and A[x] « B[x].

Some sets of Indices that will appear are

Z = {..., -2, -1, 0, 1, 2, ...}

N = {o, 1, 2, ...}

N' = {l, 2, 3, ...};

the letter I will always denote an index taking the values 0 and 1. An

unmarked summation sign (£) indicates that summation takes place as re-

peated Indices range over permissible values.

1.2 Sobolev Modules. The space of testing functions & consists of the

functions c p on the real line (= R) which have partial derivatives of all