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Normed spaces 1. In a topological linear space, the linear structure is compatible with the topology. This compatibility is more at hand if the topology is induced by a norm. A norm on a linear space X is a real-valued functional, usually denoted by II II, which verifies the conditions (x, y E X, a ElK): 1) Ilxll ;;::: 0; Ilxll = 0 if and if x = 0; 2) Ilx + yll ~ Ilxll 3) Ilaxll + Ilyll (the triangle inequality); = lal . Ilxll· A linear space X with a given norm II lion it is called a normed space and is denoted by (X, II II).

D + (3A(x) a,(3 E K A linear operator A is bounded if IIAxl1 ~ M < 00 for all X E D(A) with Ilxll = 1. In this case, the (finite) number sup IIAxl1 is xED(A),lIxll=1 called the norm of the operator A and is denoted by IIAII. 1. A linear operator is continuous if and only if it is bounded. EXAMPLES. 1. 1) with any of the norm (1)- (3), and a rectangular matrix (aik) of order n x m, i E {1, ... , m}, k E {1, ... . , n}, aik E C. The equalities n Yi = l:::aikxk, i = 1, ... ,m k=l define a linear continuous operator A: en ~ em (y = Ax for x = (Xl'"'' Xn) E E en, Y = (Yl, ...

5) show that the spaces L(X1' X2; a linear operator which maps Xl sup IIGx,ll. IlxIil :::;: 1 ~) and L(X1,L(X2'~)) are linearly isometric. Then I Gil :( E Similar results are true for n ~ 2. Let 1 :( p :( n -1 be fixed. The spaces Ln (X 1, ... , Xn; ~) and Lp(X1' ... ,Xp; L n- p(Xp+1, ... 2 (on duality). Due to the symmetric role of the spaces Xi, i for any arbitrary rearrangements of the indexes. = 1, ... ,n, a similar result holds DIFFERENTIAL CALCULUS IN NORMED SPACES 50 We consider now the particular case Xl = X2 = ...