[keithalewis]

The Wikipedia page on this is sufficient. If F:X -> Y is a function between normed linear spaces then DF:X -> L(X,Y), where L(X,Y) is the vector space of linear operators from X to Y, satisfies F(x + h) = F(x) + DF(x)h + o(h). A function is differentiable if it can be locally approximated by a linear operator.

Some confusion arises from the difference between f:R -> R and f':R -> R. It's Fréchet derivative is Df:R -> L(R,R) where Df(x)h = f'(x)h. Row vectors and column vectors a just a clumsy way of thinking about this.

BTW, all you need in order to publish on arixv.org is to know a FoF. There is no rigorous peer review. https://arxiv.org/abs/1912.01091, https://arxiv.org/abs/2009.10852.