Prove that norm vector $\lVert \vec{u}\rVert$ is equal zero then vector $\vec{u}$ is equal zero

Question

How to prove this without using algebraism?

Prove that $\lVert\vec{u}\rVert=0 \iff \vec{u}=\vec{0}$.

Question 1.8 of Geometria Analítica Third Ed., Ivan Camargo & Paulo Boulos

Answer 1

Recalling that, by definition, norms are both:

• positive definite (i.e. for all $\vec{x}$, $\lVert \vec{x}\rVert = 0 \implies \vec{x} = \vec{0}$)
• homogeneous (i.e. for all scalars $s$ and vectors $\vec{x}$, $\lVert s\vec{x} \rVert = s\lVert \vec{x} \rVert$)
• subadditive (not needed here)

We prove the two implications, starting with $\implies$:

$\lVert\vec{u}\rVert = 0$ implies $\vec{u} = \vec{0}$ by positive definiteness.

Moving on to $\impliedby$:

$\vec{u} = \vec{0}$ implies $\vec{u} = 0\vec{u}$ and thus $\lVert\vec{u}\rVert = \lVert 0\vec{u}\rVert$ and $\lVert 0\vec{u}\rVert = 0\lVert \vec{u}\rVert = 0$ by homogeneity.

Note that sometimes in the definition of norm positive definitness is, in fact, replaced by $\lVert \vec{u} \rVert = 0 \iff \vec{u} = \vec{0}$, since it is equivalent and a very commonly used property of norms.

Hope this helps!