I am familiar with the $\ell^p$ spaces, but recently I've come across the following notion without explanation. I looked it up on Wikipedia but I could not find the definition either.
Let $(X,d)$ be a (pointed) metric space. What is meant by $\ell^p(X)$ (here $p \in [1,\infty])$.
I suspect that $\ell^{p}(X)$ is the set of sequences $(x_{n})$ in $X$ such that $$\left(\sum^{\infty}_{n=1}d(b,x)^{p}\right)^{\frac{1}{p}}<\infty$$ where $b\in X$ is the special point of the pointed metric space $(X,d)$. (Note that $\ell^{p}(X)$ is dependent on $b$.)