Positive Scalar Multiple of Norm on Vector Space is Norm

Theorem
Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.

Let $\alpha > 0$ be a real number.

Define $\norm {\, \cdot \,}' : X \to \R_{\ge 0}$ by:


 * $\norm x' = \alpha \norm x$

for each $x \in X$.

Then $\norm {\, \cdot \,}'$ is a norm on $X$.

Suppose that $x \in X$ is such that:


 * $\norm x' = 0$

Then we have:


 * $\alpha \norm x = 0$

Since $\alpha > 0$, it follows that:


 * $\norm x = 0$

From, we have $x = {\mathbf 0}_X$.

Let $\lambda \in \GF$ and $x \in X$.

Then we have:

Let $x, y \in X$.

Then we have: