Topology of $p$-integrable functions space.

Question

In a reference, I read that topology of $L^p(\mathbb{R}^n)$, $1\leq p\leq \infty$. What is the topology of $L^p(\mathbb{R}^n)$? I know that $f\in L^p$, $\|f\|_{p}^{p}=\int_{\mathbb{R}^n}|f(x)|^{p}dx$. Then, the topology is $\tau=\left\{B(f,r):f\in L^p, r>0\right\}$?

Answer 1

Generally speaking, any metric space $(X,d)$ has a topological structure. The natural topology (associated to the distance $d$) is the set of all 'open' sets for distance $d$, i.e. the set

$$ \mathcal U = \{ A \subset X : \forall y \in A, \ \exists \varepsilon > 0, \ B(y,\varepsilon) \subset A\} $$

where $B(y, \varepsilon) = \{ x \in X : d(x,y) < \varepsilon\}$.

You can quite easily show that $\mathcal U$ is indeed a topology on $X$.

Of course, this applies to normed vector spaces, since to any norm $\| \cdot \|$ you can associate the natural distance $d : (x,y) \mapsto \|x-y\|$.

So in your case, you can immediately see how the natural topology is defined on $L^p(\mathbb R^n)$ equipped with the $L^p$ norm.

Note that the topology associated to a normed vector space WILL depend on the norm you choose on that space. However, it can be shown that the norm you choose will not matter if you vector space is of finite dimension (this is a direct consequence of the fact that all norms are equivalent in a finite dimension vector space). Nonetheless, $L^p(\mathbb R^n)$ is obviously not a finite dimension vector space so this does not hold in your case.