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G Y,at >Q,iE ^ {1, . . ,n}> ; Y is balanced if and only if Y — balY, where balY = {ay;y£Y, \a\ < 1}. 8 Let (X, ||-||) be a real normed space and D (A, B) be a certain distance between the subsets A,B C X (D will be specified later). For Y C X, we call: (i) defect of convexity of Y (with respect to D), the quantity dcoNV (D) (Y) = D (Y, convY); (ii) defect of linearity of Y, the quantity dLiN(D)(Y)=D{Y,spanY); (Hi) defect of balancing of Y, the quantity dBAL(D)(Y) = D(Y,balY); (iv) defect of absorption of bounded Y, the quantity dABS (D) (Y) = 0, if Y is absorbent and dABs{D){Y) = D(Y,B{0,RY)) if Y is not absorbent, where Ry B(0,RY) = = sup{||y||;j,eY}, {xeX;\\x\\

But if / is not, for example, monotone on [0,1], then it is n a t u r a l to ask how much of its degree of monotonicity is preserved by the Bernstein polynomials Bn ( / ) (x). 7. 6 (f) ([0,1]), Vn G N . O n C h a p t e r 6: D e f e c t o f P r o p e r t y i n F u n c t i o n a l A n a l y s i s This chapter introduces and studies some defects of property in functional analysis: defect of orthogonality, defect of convexity, of linearity, of balancing for sets, defect of subadditivity (additivity), of convexity for functionals, defect of symmetry, of normality, of idempotency, of permutability for linear operators, defect of fixed point.

239). Let / : E —» R. For XQ G E, the quantity defined by u (x0; f) = inf {6 [f (V n E)]; V e V (x0)} , is called oscillation of/ at XQ, where V (x0) denotes the class of all neighborhoods of x0 and S [A] = sup {|ai — a2\;ai,a2 G A] represents the diameter On Chapter 5: Defect of Property in Real Function Theory 25 of the set 4 c R . For XQ limit point of E, the quantity defined by Q (*„;/) = inf {S[f(VnE\ {x0})] ;VeV (x0)} is called pointed oscillation of / a t XQ. For e > 0, the quantity defined by w ( / ; e ) B = s u p { | / ( a r i ) - f (x2)\;\xl - x2\ 0.

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