  Stochastic gradient methods have been a popular and powerful choice of
optimization methods, aimed at minimizing functions. Their advantage lies in
the fact that that one approximates the gradient as opposed to using the full
Jacobian matrix. One research direction, related to this, has been on the
application to infinite-dimensional problems, where one may naturally have a
Hilbert space framework. However, there has been limited work done on
considering this in a more general setup, such as where the natural framework
is that of a Banach space. This article aims to address this by the
introduction of a novel stochastic method, the stochastic steepest descent
method (SSD). The SSD will follow the spirit of stochastic gradient descent,
which utilizes Riesz representation to identify gradients and derivatives. Our
choice for using such a method is that it naturally allows one to adopt a
Banach space setting, for which recent applications have exploited the benefit
of this, such as in PDE-constrained shape optimization. We provide a
convergence theory related to this under mild assumptions. Furthermore, we
demonstrate the performance of this method on a couple of numerical
applications, namely a $p$-Laplacian and an optimal control problem. Our
assumptions are verified in these applications.
