Fundamental bounded resolutions and quasi-$(DF)$-spaces

We introduce a new class of locally convex spaces $E$, under the name quasi-$(DF)$-spaces, containing strictly the class of $(DF)$-spaces. A locally convex space $E$ is called a quasi-$(DF)$-space if (i) $E$ admits a fundamental bounded resolution, i.e. an $\mathbb{N}^{\mathbb{N}}$-increasing family of bounded sets in $E$ which swallows all bounded set in $E$, and (ii) $E$ belongs to the class $\mathfrak{G}$ (in sense of Cascales--Orihuela). The class of quasi-$(DF)$-spaces is closed under taking subspaces, countable direct sums and countable products. Every regular $(LM)$-space (particularly, every metrizable locally convex space) and its strong dual are quasi-$(DF)$-spaces. We prove that $C_{p}(X)$ has a fundamental bounded resolution iff $C_{p}(X)$ is a quasi-$(DF)$-space iff the strong dual of $C_{p}(X)$ is a quasi-$(DF)$-space iff $X$ is countable. If $X$ is a metrizable space, then $C_k(X)$ is a quasi-$(DF)$-space iff $X$ is a Polish $\sigma$-compact space. We provide numerous concrete examples which in particular clarify differences between $(DF)$-spaces and quasi-$(DF)$-spaces.