Geodesic
On the Dimension of a Complete Metrizable Topological Vector Space
Canadian mathematical bulletin, Tome 20 (1977) no. 2, pp. 271-272

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The purpose of this note is to prove a result which is known to hold for Fréchet spaces [1, Chapitre II, §5, Exercise 24]. M. M. Day [2, p. 37] attributes the Banach space case to H. Löwig, although the earliest version that we have been able to find is that given by G. W. Mackey in [7, Theorem 1-1]. Recently H. E. Lacey has given an elegant proof for Banach spaces [5]. It is perhaps interesting to note that the non-locally convex case can be deduced from these known results which are established by duality arguments. 
Popoola, J. O.; Tweddle, I. On the Dimension of a Complete Metrizable Topological Vector Space. Canadian mathematical bulletin, Tome 20 (1977) no. 2, pp. 271-272. doi: 10.4153/CMB-1977-042-x
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