Uncountably infinite algebraic genericity and spaceability for sequence spaces

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.