Example

— 1) In the normed space $\mathscr{C}(I; \mathbf{C})$ (on the field $\mathbf{C}$) of functions, continuous on $I = \{0, 1\}$, with values in $\mathbf{C}$, the restrictions to $I$ of the functions $x^n \ (n \in \mathbf{N})$ form a total set, by the Weierstrass-Stone theorem (GT, X, § 4.2, th. 3). Similarly, the restrictions to $I$ of the functions $e^{2n\pi ix} \ (n \in \mathbf{Z})$ form a total set (GT, X, § 4.4, prop. 8), in the subspace $P$ of $\mathscr{C}(I, \mathbf{C})$ formed of functions such that $f(0) = f(1)$.

2) Every absorbent set in a topological vector space $E$ over a non-discrete valued division ring (and in particular every neighbourhood of 0 in $E$) is a total set since it generates $E$ (I, p. 7). Thus a linear variety that is not dense in $E$ is necessarily a nowhere dense set in $E$ (GT, IX, § 5.1) since its closure cannot contain an interior point.