Abstract
Suppose
p
n
be sequence of positive reals. By
H
w
p
n
, we represent the space of all formal power series
∑
n
=
0
∞
a
n
z
n
equipped with
∑
n
=
0
∞
λ
a
n
/
n
+
1
p
n
<
∞
, for some
λ
>
0
.
Various topological and geometric behavior of
H
w
p
n
and the prequasi ideal constructs by
s
-numbers and
H
w
p
n
have been considered. The upper bounds for
s
-numbers of infinite series of the weighted
n
-th power forward shift operator on
H
w
p
n
with applications to some entire functions are granted. Moreover, we investigate an extrapolation of Caristi’s fixed point theorem in
H
w
p
n
.