CHAPTER l

Function-Analytic Preliminaries

1. The term "Banach space" will always denote a Banach space over the reals

except in Appendix A. For definition and elementary properties of a Banach

space the reader is referred to [18].

2. Let £ b e a Banach space with norm || • || and zero element 0. If S is a

subset of E, then its closure and boundary will be denoted by £ and dS resp.,

and the empty set by 0. J denotes the identity map E onto E. By "subspace"

of E we always mean "linear subspace." If XQ is a point of E and p a positive

number, the set {x G E \ ||x - xoll p} is called the ball with center xo and

radius p. It will be denoted by B(XQ, p).

3. DEFINITION, (i) Let E\ and E2 be closed subspaces of E and suppose

every point x G E can be represented uniquely as

x = xi + X2, xi G £?i, X2 G #2 . (1)

Then we write

E = £i+£

2

(2)

and say: E is the direct sum of E\ and £?2- Each of the spaces E\ and £2 is said

to be a direct summand of £", and E\ and E2 are said to be complementary to

each other.

We note that the uniqueness of the representation (1) is equivalent to saying

that

£1 n £

2

= 0, (3)

where the symbol "fl" denotes intersection.

(ii) If E\ and E2 are Banach spaces, then the "product" E\ x E2 is defined

as the space of couples (xi, X2), Xi E 2£i, X2 G E2 with the linear operations

(xi, x2) + (2/1,2/2) = (&i +VUX2 + 2/2),

A(xi,X2) = (Axi,X2), A real.

With a proper norm, E\ x £2 becomes a Banach space (see [18, p. 89]), and

by identifying the points (xi, 0) and (0, X2) of E\ x E2 with the points x\ of E\

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http://dx.doi.org/10.1090/surv/023/02