# Function spaces

Suppose that $\mathbb{T} \subset [0,\infty)$ and $\sup \mathbb{T}=\infty$. Define the following spaces of continuous functions $$L_{\infty} = \{ f \colon \mathbb{T} \rightarrow \mathbb{R} | \sup_{t \in \mathbb{T}} |f(t)| < \infty \},$$ $$C=\{f \colon \mathbb{T} \rightarrow \mathbb{R} | \displaystyle\lim_{t \rightarrow \infty} f(t) < \infty \},$$ $$C_0 = \{f \colon \mathbb{T} \rightarrow \mathbb{R} | \displaystyle\lim_{t \rightarrow \infty} f(t)=0 \},$$ all equipped with the norm $$||f||_{\infty} = \sup_{t \in \mathbb{T}} |f(t)|.$$ Let $K = \{f \colon \mathbb{T} \rightarrow \mathbb{R} | f$ is $\Delta$-differentiable on $\mathbb{T}^{\kappa}\}$. Define $$L_{\infty}(\Delta) = \{f | f \in K, f^{\Delta} \in L_{\infty} \},$$ $$C(\Delta) = \{ f | f \in K, f^{\Delta} \in C\},$$ $$C_0(\Delta) = \{f | f\in K, f^{\Delta} \in C_0\}.$$ These spaces equipped with the norm $$||f||_{\Delta} = |f(0)|+||f^{\Delta}||_{\infty}$$ form Banach spaces.