Bounded Function Space

Last updated Nov 6, 2022

Let $X$ be a Set and let $(Y, ||\cdot||)$ be a Normed Vector Space. Then $$B(X, Y) := {f: X \to Y : \sup\limits_{x \in X} ||f(x)|| < \infty}$$ is a Normed Vector Space with the Supremum Norm $||\cdot||_{\infty}$.

This follows from noting that the Supremum Norm is an Extended Norm, Functions to a Vector Space form a Vector Space and Restricting an Extended Norm to elements of Finite Norm form a Normed Vector Space. $\square$