The gradient of a function on a Banach space is an element of the dual space

Question

Can somebody explain me why gradient descent in Banach space does not make sense? As pointed out by Sebastien Bubek in his blog, the gradient is an element of the dual space $\mathcal{B}^*$. But I cannot see this.

Answer 1

What is the gradient $\nabla f$, to begin with? It is an object that we can use to find the directional derivatives of $f$. More precisely, it is a linear map whose argument is a vector $u\in B$ and the output is the directional derivative $D_u f$ along that vector.

So, $\nabla f$ is a linear map from space $B$ to real numbers (or complex if you consider complex scalars). This makes it an element of the dual space $B^*$.