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By Jose Luis Massera, Juan F. Schaffer

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We first show that Z n V = (0). If x E (2 n V )\ {0}we have which is absurd. 13. THECLASS OF SUBSPACES OF A We next show that Z be given. Then +V BANACHSPACE 23 is closed. D, 2 V is closed. V . Assume that this is not the case. We now claim that Y C Z Since Z V is a subspace, Y n (2 V ) is a proper subspace of Y ; for every X > 1 there exists (cf. E) an element y E Y \ (0) such that 11 y 11 X d( Y n ( Z V ) ,y ) . There exists, then, z E Z such that 11 y - z 11 X 6’( Y , 2)I/ y 11. But (I - P)z E Y , (I - P ) x = z - PZ E Z+ V , so that + + < + + < + II Y II < 4 y n (Z V ) , Y ) < IIY - (1 - P)z II < I/ I - p II II Y 3 II < x2 II I - p II a‘( y , Z ) II y II.

The essential limits for t -+ - co or t -+ Function spaces will be denoted by bold-faced capitals such as F, G , and classes of function spaces by script capitals, such as N, 9; either standing alone or with subscripts, etc. I n the case of L(X) (defined in the preceding subsection, and more precisely in the next subsection but one), as well as in the cases discussed in Sections 22, 23, and 24, this notational convention reflects a deeper underlying structure: as the precise definitions will show, in these cases the symbol without the argument, say F, indicates a space of real-valued functions, and F(X) is a related space of functions with values in X .

If Y # (0) is a subspace and x E X \ Y , then y [ Y , x] = 2 sin &+( Y , x), and 11 x 11-l d( Y , x) = sin +( Y , x) (thus 11 x 11 < h d( Y , x), 28 Ch. 1. GEOMETRY OF BANACH SPACES > x # 0 is equivalent to sin +(Y, x ) k l ) . If Y, Z # (0) are subspaces with Y n Z = (0}, then y[Y, 21 = 2 sin 8 +( Y, 2). Proof. Immediate from the definitions. B. If ( Y , 2) is a disjoint closed dihedron, not ( X , (0))nor ((01, X ) , and P is the projection along Y onto 2, then sin Y, 2) = 11 P 11-l. a( Proof.

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