arXiv subjects
Publications and source records attributed to Matěj Novotný.
We show that the basis constant of every retractional Schauder basis on the Free space of a graph circle increases with the radius. As a consequence, there exists a uniformly discrete subset $M\subset\mathbb{R}^2$ such that $\mathcal F(M)$ does not have a retractional Schauder basis. Furthermore, we show that for any net $ N\subseteq\mathbb{R}^n$ there is no retractional unconditional basis on the Free space $\mathcal F(N)$.
We give several structural results concerning the Lipschitz-free spaces $\mathcal F(M)$, where $M$ is a metric space. We show that $\mathcal F(M)$ contains a complemented copy of $\ell_1(Γ)$, where $Γ=\text{dens}(M)$. If $\mathcal N$ is the net in a finite dimensional Banach space $X$, we show that $\mathcal F(\mathcal N)$ is isomorphic to its square. If $X$ contains a complemented copy of $\ell_p, c_0$ then $\mathcal F(\mathcal N)$ is isomorphic to its $\ell_1$-sum. Finally, we prove that for all $X\cong C(K)$ spaces $\mathcal F(\mathcal N)$ are mutually isomorphic spaces with a Schauder basis.