# Is the function space in the topological space level?

I am learning $$L^p$$ space, whose definition is based on function spaces.

In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space have a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space.

cited from wiki

according to the description, the function space is in the topological space level, right? • In short: No, $L^p$ is at the level of a normed space (and, Yes, therefore a topological space with the topology induced by the norm). For the precise definition of $L^p$ one needs a bit of measure theory. Sep 18 '19 at 15:50
• No, it is a normed vector space. The norm of $f$ is the $L^p$ norm $$\left(\int \|f\|^p\right)^{1/p}$$ Sep 18 '19 at 15:53
• In addition, $L^2$ is a inner product space (in fact, Hilbert). Sep 18 '19 at 16:02
• But a function space in general doesn't even have a topology; it's just a vector space, if the codomain is a vector space. Sep 18 '19 at 17:27
• By the definition given, function spaces need not be topological, nor vector spaces. Therefore, their Venn diagram region would cut through every one of your sets, but also include a lot that is outside. Function spaces are far more common than you realize. Almost every explicitly-defined vector space you will encounter can be thought of as a function space. For example, $\Bbb R^n$ is the space of all functions from $\{1, 2, 3, ..., n\} \to \Bbb R$. Sep 19 '19 at 2:09