Bounded generators of linear spaces

Takashi Ito and Thomas I. Seidman

Vol. 26 (1968), No. 2, 283–287

DOI: 10.2140/pjm.1968.26.283

Abstract

Let S_φ = {x ∈ X : sup_αφ_α(x) < ∞} where φ = {φ_α} is a family of semi-norms determining the topology of X. It is shown that φ may be chosen so S_φ 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 S_φ is dense for Fréchet spaces.

Mathematical Subject Classification

Primary: 46.01

Milestones

Received: 26 July 1967

Published: 1 August 1968