Bounded Function Spaces

Subjects: Metric and Normed Spaces
Links: Normed Vector Spaces, Metric Spaces, Complete Metric Spaces

A function $f:S\to X$ is bounded if there are $c\in \mathbb{R}$ and $x_0\in X$ such that, for any $z\in S$ $$ d(f(z), x_0)\le c $$We denote

and we define $$ d_\infty(f,g) = \sup_{z \in S}d(f(z), g(z)) $$$d_{\mathrm{\infty }}$ is a metric on $\mathcal{B}(S,X)$. This metric is called the uniform metric.

Prop: If $X$ is a complete metric space, then $\mathcal{B}(S,X)$ is also complete with the uniform metric.

If $V$ is a vector space, the set of all function from $S$ to $V$ is a vector space with the operations $$ (f+g)(z) := f(z)+g(z) \qquad (\lambda f)(z) := \lambda f(z) $$
If $V$ is a normed space with norm $\parallel \cdot \parallel$, then $\mathcal{B}(S,V)$ is a vector space and$$ |f|\infty := \sup{z \in S}|f(z)| $$is a norm in $\mathcal{B}(S,V)$. This norm is called the uniform norm.