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The space $\mathrm{\mathscr{H}𝒦}$ of Henstock-Kurzweil integrable functions on $[a,b]$ is the uncountable union of Fréchet spaces $\mathrm{\mathscr{H}𝒦}\left(X\right)$. In this paper, on each Fréchet space $\mathrm{\mathscr{H}𝒦}\left(X\right)$, an $F$-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space. It is known that every control-convergent sequence in the $\mathrm{\mathscr{H}𝒦}$ space always belongs to a $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space for some $X$. We illustrate how to apply results...

In 1938, L. C. Young proved that the Moore-Pollard-Stieltjes integral ${\int }_a^{b}f\mathrm{d}g$ exists if $f\in {\mathrm{B}V}_{\phi }[a,b]$, $g\in {\mathrm{B}V}_{\psi }[a,b]$ and ${\sum }_{n=1}^{\infty }{\phi }^{-1}(1/n){\psi }^{-1}(1/n)<\infty$. In this note we use the Henstock-Kurzweil approach to handle the above integral defined by Young.