In this paper, we deal with the first and second widths of the real projective space
R
P
n
\mathbb {RP}^{n}
, for
n
n
ranging from
4
4
to
7
7
, and for this we used some tools from the Almgren-Pitts min-max theory. In a recent paper, Ramirez-Luna computed the first width of the real projective spaces, and, at the same time, we obtained optimal sweepouts realizing the first and second widths of those spaces.