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 an $\mathbb{F}$-linear subspace, equipped with the restricted norm $\|\cdot\|_Y:Y\to[0,\infty)$ defined by $\|y\|_Y=\|y\|_V$ for every $y\in Y$. If $(Y,\|\cdot\|_Y)$ is a Banach space, then $Y$ is closed in $V$ with respect to the norm topology induced by $\|\cdot\|_V$.