Remarks on independent sequences and dimension in topological linear spaces

We show that for a metrizable locally convex space $X$ the following conditions are equivalent: (i) every linearly independent sequence in $X$ has an $\omega$-independent subsequence; (ii) $X$ contains no subspace isomorphic to $\varphi$; (iii)$X$ admits a continuous norm. We also show that a dual Banach space equipped with the weak$^\ast$ topology satisfies (i). Moreover, we are concerned with the algebraic dimension of closed convex subsets of $F$-spaces