Uncountably infinite algebraic genericity and spaceability for sequence spaces

TL;DR

This paper establishes conditions under which the complement of a subset in a topological vector space contains uncountably many dense and closed infinite-dimensional subspaces, with applications to classical sequence spaces.

Contribution

It introduces new criteria for spaceability and algebraic genericity in sequence spaces, extending previous results to uncountably infinite collections.

Findings

Conditions for uncountably many dense subspaces

Conditions for uncountably many closed infinite-dimensional subspaces

Applications to classical sequence spaces like ll^p

Abstract

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.