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By Goursat E.

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Extra info for A course in mathematical analysis. - part.2 Differential equations

Sample text

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.

In the sense of Bourbaki [l] (p. 48). , T h e most conspicuous instance of a coupled pair is of course that of an arbitrary Banach space X and its dual X * , coupled by the natural or “evaluation” functional (x, x*) = x*(x). Wherever the pair X , X* shall occur in either order, the coupling will be understood to be by means of this functional. 2) means that W X ‘ is a subspace of characteristic 1 of X * , in the terminology of Dixmier [I] (a duxial subspace of X * , according to Ruston [l]).

B); hence a dihedron (Y, 2) is (X, X * ) disjoint if and only if it is disjoint and closed. 13. The class of subspaces of a Banach space Two mettics In this section we give an account of two ways in which a topological and metrical structure may be imposed on the class of all subspaces of a Banach space, and of those properties of these structures which are relevant for our purpose. We shall follow in the main the exposition of Berkson [l], to which we refer the reader for details. T h e contents of this entire section will be applied for the first time in Chapter 7.