Normed Spaces

Definition (Norm, Normed Space)

A norm on a real or complex vector space $X$ is a real-valued function

$$\Vert \cdot \Vert :X\to \mathbb{R}$$

having the following properties:
(N1) Positivity. $\Vert x\Vert >0$ whenever $x\ne 0$
(N2) Homogenity. $\Vert \alpha x\Vert =|\alpha |\Vert x\Vert$ for all scalars $a$
(N3) Subadditivity. $\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert$

An normed vector space (or normed linear space or simply normed space) is a pair $(X,\Vert \cdot \Vert )$ consisting of a real or complex vector space $X$ and a norm $\Vert \cdot \Vert$ on $X$.

Setting $a=0$ in (N2) yields $\Vert 0\Vert =0$.

Proposition (Metric Induced by a Norm)

If $\Vert \cdot \Vert$ is a norm on a real or complex vector space $X$, then

$$d(x,y)=\Vert x-y\Vert \forall x,y\in X$$

defines a metric $d$ on $X$.

Definition (Norm Topology)

If $(X,\Vert \cdot \Vert )$ is a normed space, then the topology on $X$ associated with the metric induced by the norm is called the norm topology.

TODO equivalent norms, subspaces, product, quotient TODO open ball, closed ball, sphere

Proposition (Continuity of Addition, Scalar Multiplication)

Let $(X,\Vert \cdot \Vert )$ be a normed space. The addition of vectors, $X\times X\to X$, $(x,y)\mapsto x+y$, and the the multiplication of vectors by scalars, $\mathbb{K}\times X\to X$, $(\alpha ,x)\mapsto \alpha x$, are continous with respect to the norm topology (where the products are equipped with the product topologies). In particular, a normed space is a topological vector space with respect to the norm topology.

Proof   This follows from the inequalities

$$\Vert (x+y)-(x_0+y_0)\Vert \le \Vert x-x_0\Vert +\Vert y-y_0\Vert$$

and

$$\Vert \alpha x-{\alpha }_0{x}_0\Vert \le |\alpha |\Vert x-x_0\Vert +|\alpha -{\alpha }_0|\Vert x_0\Vert .$$

fix $\square$

Proposition (Uniform Continuity of the Norm)

Let $(X,\Vert \cdot \Vert )$ be a normed space. Then we have

$$|\Vert x\Vert -\Vert y\Vert |\le \Vert x-y\Vert \forall x,y\in X.$$

Consequently, the norm is a uniformly continous function with respect to the norm topology.

Proposition (Quotient Norm)

Let $Y$ be a closed linear subspace of a normed space $(X,\Vert \cdot \Vert )$. Then

$$\Vert x+Y\Vert =\underset{y\in Y}{\inf }\Vert x+y\Vert \forall x\in X$$

defines a norm on the quotient vector space $X/Y$.

Duals of Normed Spaces