Let $(V,\|\cdot\|_V)$ be a [Banach space](/page/Banach%20Space) over $\mathbb{F}$, where $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$. Let $Y\subset V$ be a linear subspace over $\mathbb{F}$ that is closed in the norm topology induced by $\|\cdot\|_V$. Define the restricted norm $\|\cdot\|_Y:Y\to[0,\infty)$ by $\|y\|_Y=\|y\|_V$ for every $y\in Y$. Then $(Y,\|\cdot\|_Y)$ is a Banach space over $\mathbb{F}$.