For a normed space, is a vector subspace of it with the restriction norm also a normed space

normed-spaces

Question. Let $X = (X,\|\cdot\|)$ be a normed space and let $S \subset X$ be a subspace of $X$. Is $S$ a normed space (with the norm $\|\cdot\|$ of $X$) ?

My intuition. All we need to say for $S$ to be a normed space is that $S$ is a vector space and that $\|\cdot\|$ defines a norm on $S$. Since $S$ is a subspace of $X$, it also verifies the axioms to be a vector space and thus, $S$ is a vector space. On the other hand, if $\|\cdot\|$ is a norm on $X$ it will surely also be a norm on $S$. Thus, the answer is affirmative.

Is my thinking correct?

Thanks for any help in advance.

Best Answer

Let $(X,\|.\|)$ be a real normed vector space and let $S\subset X$ a subspace of $X$.

  1. $S$ is a vector space(in particular $0\in S)$;
  2. Let $$\|.\|_S:S\to \Bbb R^+$$ $$x\mapsto \|x\|$$

Then

a) $\forall \lambda \in \Bbb R,\forall x\in S ,\|\lambda x\|_S=\|\lambda x\|=|\lambda|\| x\|=|\lambda|\| x\|_S$;

b) $\| 0\|_S=\| 0\|=0;$

c) $\forall x\in S, $ if $\| x\|_S=0$, then $\| x\|=0$, then $x=0$;

d) $\forall x,y \in S, \| x+y\|_S=\| x+y\|$ and $\| x+y\|\leq\| x\|+\| y\|$, i.e. $\| x+y\|_S\leq\| x\|_S+\| y\|_S$

Thus, $(S,\|.\|_S)$ is a real normed vector space.

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