Extending a result of Mashreghi and Ransford, we prove that every complex separable infinite-dimensional Fréchet space with a continuous norm is isomorphic to a space continuously included in a space of holomorphic functions on the unit disc or the complex plane, which contains the polynomials as a dense subspace. As a consequence, we deduce the existence of nuclear Fréchet spaces of holomorphic functions without the bounded approximation.

Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Fréchet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space $\varphi $ of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on $\varphi$). (2) The space of all test functions for distributions, which is also a complete direct sum of Fréchet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Fréchet space contains a dense hyperplane that admits no transitive operator.

We introduce the concept of logarithmic dimension of a compact set. In terms of this magnitude, the extension property and the diametral dimension of spaces $\varepsilon \left( K \right)$ can be described for Cantor-type compact sets.

The problems on isomorphic classification and quasiequivalence of bases are studied for the class of mixed $F-$, $\text{DF}$-power series spaces, i.e. the spaces of the following kind

$$G(\lambda ,a)=\underset{p\to \infty }{\mathop{\lim }}\,\text{proj}\left( \underset{q\to \infty }{\mathop{\lim }}\,\text{ind}\left( {{\ell }_{1}}\left( {{a}_{i}}\left( p,q \right) \right) \right) \right),$$

where ${{a}_{i}}(p,\,q)\,=\,\exp \left( \left( p\,-\,{{\lambda }_{i}}q \right){{a}_{i}} \right),\,p,\,q\,\in \,\mathbb{N},\,\text{and}\,\lambda \,\text{=}\,{{\left( {{\lambda }_{i}} \right)}_{i\in \mathbb{N}}},\,a=\,{{({{a}_{i}})}_{i\in \mathbb{N}}}$ are some sequences of positive numbers. These spaces, up to isomorphisms, are basis subspaces of tensor products of power series spaces of $F-$ and $\text{DF}$-types, respectively. The m-rectangle characteristic $\mu _{m}^{\lambda ,a}\left( \delta ,\,\varepsilon ;\,\tau ,\,t \right),\,m\,\in \,\mathbb{N}$ of the space $G\left( \text{ }\!\!\lambda\!\!\text{ ,}\,a \right)$ is defined as the number of members of the sequence ${{({{\lambda }_{i}},{{a}_{i}})}_{i\in \mathbb{N}}}$ which are contained in the union of m rectangles ${{P}_{k}}\,=\,\left( {{\delta }_{k}},\,{{\varepsilon }_{k}} \right]\,\times \,\left( {{\tau }_{k}},\,{{t}_{k}} \right]$ , $k\,=\,1,2,\ldots ,m$. It is shown that each m-rectangle characteristic is an invariant on the considered class under some proper definition of an equivalency relation. The main tool are new compound invariants, which combine some version of the classical approximative dimensions (Kolmogorov, Pełczynski) with appropriate geometrical and interpolational operations under neighborhoods of the origin (taken from a given basis).