It is proved that if a metrizable locally convex space is not nuclear, then it does not satisfy the Lévy-Steinitz theorem on rearrangement of series.

Suppose that $X$ is a Fréchet space, $\langle a_{ij}\rangle$ is a regular method of summability and $\left(x_i\right)$ is a bounded sequence in $X$. We prove that there exists a subsequence $\left(y_i\right)$ of $\left(x_i\right)$ such that: either (a) all the subsequences of $\left(y_i\right)$ are summable to a common limit with respect to $\langle a_{ij}\rangle$; or (b) no subsequence of $\left(y_i\right)$ is summable with respect to $\langle a_{ij}\rangle$. This result generalizes the Erdös-Magidor theorem which refers to summability of bounded sequences in Banach spaces. We also show that two analogous results for some ${\omega }_1$-locally convex spaces...

We give a representation of the spaces $C^{\infty }\left({ℝ}^N\right)\cap H^{k,p}\left({ℝ}^N\right)$ as spaces of vector-valued sequences and use it to investigate their topological properties and isomorphic classification. In particular, it is proved that $C^{\infty }\left({ℝ}^N\right)\cap H^{k,2}\left({ℝ}^N\right)$ is isomorphic to the sequence space $s^{\mathbb{N}}\cap l^2\left(l^2\right)$, thereby showing that the isomorphy class does not depend on the dimension N if p=2.