On convergent sequences in locally convex topological vector spaces

Question

Assume that a sequence $(x_n)_{n\in\omega}$ of points of a locally convex topological vector space converges to zero. Is it always possible to find increasing number sequences $(n_k)_{k\in\omega}$ and $(m_k)_{k\in\omega}$ such that the sequence $(m_kx_{n_k})_{k\in\omega}$ still converges to zero?

Added in Edit. So we already know that this property (called the Mackey convergence condition) does not hold in any locally convex space. But we can ask another

Problem. Assume that a locally convex space $X$ admits an indexed family $(B_\alpha)_{\alpha\in\omega^\omega}$ of bounded sets such that (i) $B_\alpha\subset B_\beta$ for all $\alpha\le \beta$ in $\omega^\omega$ and (ii) each bounded subset $B\subset X$ is contained in some $B_\alpha$, $\alpha\in\omega^\omega$.

Does $X$ satisfy the Mackey convergence condition?

Answer 1

This is satisfied for locally convex spaces with the so-called Mackey convergence condition, i.e., for every null sequence $x$ there is a bounded absolutely convex set $B$ such that $x$ tends to $0$ in the linear hull of $B$ endowed with the Minkowski functional of $B$ as a norm. This property was introduced by Grothendieck and is satisfied by metrizable spaces and virtually all non-Banach spaces arising in analysis (test function spaces $\mathscr D(\Omega)$, distributions $\mathscr D'(\Omega)$, spaces of germs of holorphic functions on compact sets, spaces of real analytic functions,...).

Answer 2

Oh, sorry! I posed this question too quickly.

A simple example is the dual space $\ell_1=c_0^*$ to the Banach space $c_0$, endpowed with the weak$^*$ topology. The sequence $(e^*_n)_{n\in\omega}$ of coordinate functionals tends to zero, but for any increasing number sequences $(m_k)_{n\in\omega}$ and $(n_k)_{k\in\omega}$ the sequence $(m_ke^*_{n_k})_{k\in\omega}$ does not converge to zero on any sequence $(x_n)_{n\in\omega}\in c_0$ such that $x_{n_k}=\frac1{\sqrt{m_k}}$ for all $k\in\omega$.