Definition:Norm (Vector Space)
This page is about the norm on a vector space. For other uses, see Definition:Norm.
Contents |
Definition
Let $\left({K, +, \circ}\right)$ be a division ring with norm $\left|{\cdot}\right|_K$.
Let $V$ be a vector space over $K$, with zero $0_V$.
A norm on $V$ is a map from $V$ to the nonnegative reals $\left\Vert{\cdot}\right\Vert: V \to \R_+ \cup \left\{{0}\right\}$ satisfying the following three properties (for all $x,y \in V$ and $\lambda \in K$):
| N1: | Positive definiteness: | $\left\Vert {x} \right\Vert = 0 \iff x = 0_V$ |
| N2: | Positive homogeneity: | $\left\Vert{\lambda x}\right\Vert = \left|{\lambda}\right|_K \times \left\Vert{x}\right\Vert$ |
| N3: | Triangle inequality: | $\left\Vert {x + y}\right\Vert \leq \left\Vert{x}\right\Vert + \left\Vert{y}\right\Vert$ |
These may be referred to as the (vector space) norm axioms.
Normed Vector Space
A normed vector space is a vector space endowed with a norm.
Notes
In the literature, it is more common to define the norm only for $K$ a subfield of the complex numbers (and consequently $\left|{\cdot}\right|_K$ the modulus function).
However, the definition given here incorporates this approach.
some links to literature be given
You can help ProofWiki by adding this information.
To discuss it in more detail, feel free to use the talk page.
If you are able to do this, then when you have done so you can remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
