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Math. Soc, (1977), (to appear). some old and some new results, Bull. 41 [6] W. W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Grundlehren der math. Wissenschaften vol. 211. Springer-Verlag, Berlin- Heidelberg-New York, 1974. [7] B. A. Efimov, On the embedding of extremally disconnected spaces into bicompacta, In: General Topology and Its Relations to Modern Analysis and Algebra III. Proc. Third Prague Topological Symposium, Prague, 1971, edited by J. Nov~k, pp. 103-107. [8] Academia Prague, Prague, 1972.

No Kolmogorov and W. M. Tichomirow: of sets in function spaces. E-entropy and E-capacity (Russian) Uspehi Mat. Nauk 14 (86) (1959), 3-86. German translation in Arbeiten zur Informationstheorie III, Berlin 1960. 33 [3] H. K~nig: Grenzordnungen von 0peratorenidealen I, II. Math. Ann. 212 (1974), 51-77. [~ G. G. Lorentz: Metric entropy and approximation. Bull. Amer. Math. Soc. 72 (1966), 903-937. [4 G. G. Lorentz: Approximation of functions. New York, 1966o [~ B. S. Mitjagin: Approximative dimension and bases in nuclear spaces.

The inclusions stated above are the best possible which can be expected. Some weaker results are proved in [12]. 4. Entropy numbers of operators in Hilbert spaces It seems to be very complicated to compute or estimate the entropy numbers of a given operator. However, we know some results concerning the related quasi-norms. Theorem 4. +.. 12) such that S(~n ) . -~ 0 , then < ~ n = 2en(S) " = (~n fn ) and ( 6+n) Proof. If ~I >= ~2 6 = 0 , then the assertion is trivial. So we assume that >n ~ ~n > 0 .

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