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

Set yk =

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).

### Elements of the Theory of Functions by Knopp K.

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