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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
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