Vector Subspace is a Normed Vector Subspace

Last updated Nov 1, 2022

Let $(V, ||\cdot||)$ be a Normed Vector Space over $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. If $W \subset V$ is a Vector Subspace, then $(W, ||\cdot|| {\big|}_{W})$ is a Normed Vector Space called a Normed Vector Subspace.

$W$ is a Vector Space by defnition of Vector Subspace. The Norm Axioms easily follow because they already hold $V$ and $W \subset V$. $\blacksquare$