  Having been unclear how to define that a domain is strictly pseudoconvex in
the infinite-dimensional setting, we develop a general theory having Banach
spaces in mind. We first focus on finite dimension and eliminate the need of
two degrees of differentiability of the boundary of a domain, since
differentiable functions are difficult to find in infinite dimension. We
introduce $\ell$-strict pseudoconvexity for $\ell\geq 1$, $1$-strict
pseudoconvexity at the boundary, $\ell$-uniform pseudoconvexity for $\ell\geq
0$ and finally strong pseudoconvexity. Defining $\ell$-strict pseudoconvexity
and $\ell$-uniform pseudoconvexity for $\ell<2$ depends on extending a notion
of strict plurisubharmonicity to cases lacking $C^2$-smoothness, first studying
it in the sense of distribution and then considering it in infinite dimension.
Examples of strictly plurisubharmonic functions as well as strongly
pseudoconvex domains are presented, which end up related to important classical
Banach spaces. Finally, some solutions to the inhomogeneous Cauchy-Riemann
equations for $\overline{\partial}$-closed $(0,1)$-forms in
infinite-dimensional domains are shown, giving new information about domains
affinely isomorphic to the ball of $\ell_1$ which appeared in the study of
strong pseudoconvexity and some more domains biholomorphically related to open
and convex domains of $\ell_1$ such as its ball.
