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By Knopp K.

Normal history: advanced numbers, linear capabilities, units and sequences, conformal mapping. distinct proofs.

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18): Definition 10. Let X be a TVS. A linear map from X into the scalar field is called linear functional. , X' = L(X, IF). If x' is a linear functional on X, we often write = x'(x) for the value of x' at x E X. If X is a semi-NLS, it follows from Proposition 8 that X' is a B-space under the norm Ilx' II = sup{ I I : IIxII s 1); this norm is called the dual norm on X'. If X and Y are semi-NLS, an is me from X into Y is a map U : X -4 Y such that IIUx - Uyll = Ilx - yll dx, y E X.

Set yk = for k E IN and y = tyk). We claim Linear Operators and Linear Functionals 48 N that y E ll. , s IIfII k=1 Co so y ell and IIfII I <_ IIfII. x = (xk) E CO, x = For xkek so k=1 C _ xkyk and xk _ k=1 k=1 Co II < IIxII- IYkI = IIxII-11Y111 k=1 which implies that IIfII s 11y111. Hence, IIfII =11y1I1 Thus, the map U : f --4 y from c' to obviously linear. 11 is an isometry and is The map U is also onto since if y = {yk} E ll, 00 ykxk defines a continuous linear functional with U(fy) = Y.

We now show that in a quasi-normed space every sequence which converges to 0 is Mackey convergent to 0. Proposition 5. Let (X, 1 I) be a quasi-normed space and 1 - 0 in X. x. -40. ij Proof: Set n1 = 0. 3 an increasing sequence of positive integers (n k} such that I JJI<_1/k and i J l/k2 for j >_ nk. For nk 5 j < nk+1' put k. = k so 1-+0. 10. Definition 6. A semi-norm on X is a function X -4R satisfying (1) IIxiI _ 0 dxEX, (ii) iitxll = It11IxII (iii) iix + y11:5 lixii + IiyII Vx, y E X (triangle inequality).

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