Vol. 26, No. 2, 1968

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Bounded generators of linear spaces

Takashi Ito and Thomas I. Seidman

Vol. 26 (1968), No. 2, 283–287

Let Sφ = {x X : supαφα(x) < ∞} where φ = {φα} is a family of semi-norms determining the topology of X. It is shown that φ may be chosen so Sφ is dense if and only if X has a bounded generating set if and only if there is a continuous norm on X. It is shown that these conditions hold for separable Fréchet spaces and for quotients of products of Banach spaces. An example is given of a Fréchet space containing no bounded generating set thus contradicting an assertion of L. Maté that Sφ is dense for Fréchet spaces.

Mathematical Subject Classification
Primary: 46.01
Received: 26 July 1967
Published: 1 August 1968
Takashi Ito
Thomas I. Seidman