# normed vector space

Let $\mathbb{F}$ be a field which is either $\mathbb{R}$ or $\mathbb{C}$. A over $\mathbb{F}$ is a pair $(V,\lVert\cdot\rVert)$ where $V$ is a vector space over $\mathbb{F}$ and $\lVert\cdot\rVert\colon V\to\mathbb{R}$ is a function such that

1. 1.

$\lVert v\rVert\geq 0$ for all $v\in V$ and $\lVert v\rVert=0$ if and only if $v=0$ in $V$ (positive definiteness)

2. 2.

$\lVert\lambda v\rVert=\lvert\lambda\rvert\lVert v\rVert$ for all $v\in V$ and all $\lambda\in\mathbb{F}$

3. 3.

$\lVert v+w\rVert\leq\lVert v\rVert+\lVert w\rVert$ for all $v,w\in V$ (the triangle inequality)

The function $\lVert\cdot\rVert$ is called a norm on $V$.

Some properties of norms:

1. 1.

If $W$ is a subspace of $V$ then $W$ can be made into a normed space by simply restricting the norm on $V$ to $W$. This is called the induced norm on $W$.

2. 2.

Any normed vector space $(V,\lVert\cdot\rVert)$ is a metric space under the metric $d\colon V\times V\to\mathbb{R}$ given by $d(u,v)=\lVert u-v\rVert$. This is called the metric induced by the norm $\lVert\cdot\rVert$.

3. 3.

It follows that any normed space is a locally convex topological vector space, in the topology induced by the metric defined above.

4. 4.

In this metric, the norm defines a continuous map from $V$ to $\mathbb{R}$ - this is an easy consequence of the triangle inequality.

5. 5.

If $(V,\langle,\rangle)$ is an inner product space, then there is a natural induced norm given by $\lVert v\rVert=\sqrt{\langle v,v\rangle}$ for all $v\in V$.

6. 6.

The norm is a convex function of its argument.

 Title normed vector space Canonical name NormedVectorSpace Date of creation 2013-03-22 12:13:45 Last modified on 2013-03-22 12:13:45 Owner rspuzio (6075) Last modified by rspuzio (6075) Numerical id 14 Author rspuzio (6075) Entry type Definition Classification msc 46B99 Synonym normed space Synonym normed linear space Related topic CauchySchwarzInequality Related topic VectorNorm Related topic PseudometricSpace Related topic MetricSpace Related topic UnitVector Related topic ProofOfGramSchmidtOrthogonalizationProcedure Related topic EveryNormedSpaceWithSchauderBasisIsSeparable Related topic EveryNormedSpaceWithSchauderBasisIsSeparable2 Related topic FrobeniusProduct Defines norm Defines metric induced by a norm Defines metric induced by the norm Defines induced norm