  Let $X$ be a topological vector space of complex-valued sequences and $Y$ be
a subset of $X$. We provide conditions for $X \setminus Y \cup \{0\}$ to
contain uncountably infinitely many linearly independent dense vector subspaces
of $X$. We also provide conditions for $X \setminus Y \cup \{0\}$ to contain
uncountably infinitely many linearly independent closed infinite-dimensional
vector subspaces of $X$. We apply these results to a chain of spaces containing
the $\ell^p$ spaces.
