Geodesic
Necessary and Sufficient Conditions for the Equality of L(f) and l 1
Canadian journal of mathematics, Tome 34 (1982) no. 2, pp. 406-410

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Introduction. Let f be a modulus, ei = (δij) and E = {ei , i = 1, 2, ...}. The L(f) spaces were created (to the best of our knowledge) by W. Ruckle in [2] in order to construct an example to answer a question of A. Wilansky. It turned out that these spaces are interesting spaces. For example lp , 0 < p ≦ 1 is an L(f) space with f(x) = xp , and every FK space contains an L(f) space [2]. A natural question is: For which f is L(f) a locally convex space? It is known that L(f) ⊆ l 1, for all f modulus (see [2]), and l 1 is the smallest locally convex FK space in which E is bounded (see [1]). Thus the question becomes: For which f does L(f) equal l 1? In this paper we characterize such f. (An FK space need not be locally convex here.) We also characterize those f for which L(f) contains a convex ball. The final result of this paper is to show that if f satisfies f(x · y) ≦ f(x) · f(y) and L(f) ≠ l 1 then L(f) contains no infinite dimensional subspace isomorphic to a Banach space. 
Deeb, Waleed. Necessary and Sufficient Conditions for the Equality of L(f) and l 1. Canadian journal of mathematics, Tome 34 (1982) no. 2, pp. 406-410. doi: 10.4153/CJM-1982-026-1
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[1] 1. Bennett, G., Some inclusion theorems for sequence spaces, Pacific J. Math. 46 (1973), 17–30. Google Scholar

[2] 2. Ruckle, W., FK spaces in which the sequence of coordinate vectors is bounded. Can. J. Math. 25 (1973), 973–978. Google Scholar

[3] 3. Stiles, W., On properties of sub spaces of lp, 0 &lt; p &lt; 1, Trans. Amer. Math. Soc. 149 (1970), 405–415. Google Scholar

[4] 4. Wilansky, A., Functional analysis (Blaisdell, New York, 1964). Google Scholar

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