Direct Product Normed Space

Last updated Nov 1, 2022

Let $(X, ||\cdot||{X}), (Y, ||\cdot||{Y})$ be Normed Vector Spaces. Then the Direct Product Vector Space $X \times Y$ equipped with a p-norm $||\cdot||{p} : X \times Y \to \mathbb{R}{\geq 0}$ defined as $$||(x, y)||_{p} := (||x||^{p} + ||y||^{p})^{\frac{1}{p}}$$ forms a Normed Vector Space. It is known as the Direct Product Normed Space.

1. Sometimes, this space is called the “$l_{p}$ sum of $X$ and $Y$” and written $X \oplus_{p} Y$.