Benjamin R

In *Introduction to Abstract Algebra* (2014) by Fine, Gaglione, Rosenberger, p.459, we have a very clear definition of the properties of the **norm** on a vector space such that *"Any inner product space $V$ is a normed linear space with |v| = $\sqrt{<v,v>}\ $ for* [all] *v $\in V$."*

Where *A* **norm** *on a vector space is a function $|\ |\ :\ V \rightarrow \mathbb{R}\ \ $satisfying:*

$(1)\ \ |v| \geq 0 \ \ for\ all\ v \in V$

$(2)\ \ |v| = 0 \ \ if\ and\ only\ if\ \ v = 0_{v}$

$(3)\ \ |tv| = |t||v|\ \ for\ \text{[all]}\ v\ \in V\ and\ t\ \in \mathbb{R}$

more info: Consider the following statement from "Pearls in Graph Theory" (Harstfield, Ringel):

"Two graphs $\text{G}_1, \text{G}_2$ with $p$ vertices are said to be $isomorphic$ if the vertices of $\text{G}_1$ and $\text{G}_2$ can be labeled with the numbers from 1 to $p$ such that whenever vertex $i$ is adjacent to vertex $j$ in $\text{G}_1$, then vertex $i$ is adjacent to vertex $j$ in $\text{G}_2$ and conversely.

Such a labeliing is the same as a one-to-one correspondence between $\text{V(G}_1)$ and $\text{V(G}_2)$ that preserves adjancency."