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---
title: Normed Spaces
parent: Functional Analysis Basics
nav_order: 1
---
# {{ page.title }}
{% definition Norm, Normed Space %}
A *norm* on a real or complex vector space $X$
is a real-valued function
$$
\norm{\cdot} : X \to \RR
$$
having the following properties: \
**(N1)** *Positivity*. $\norm{x} > 0$ whenever $x \ne 0$ \
**(N2)** *Homogenity*. $\norm{\alpha x} = \abs{\alpha} \norm{x}$ for all scalars $a$ \
**(N3)** *Subadditivity*. $\norm{x+y} \le \norm{x} + \norm{y}$
An *normed vector space* (or *normed linear space* or simply *normed space*) is a pair $(X,\norm{\cdot})$
consisting of a real or complex vector space $X$
and a norm $\norm{\cdot}$ on $X$.
{% enddefinition %}
Setting $a=0$ in **(N2)** yields $\norm{0} = 0$.
{% proposition Metric Induced by a Norm %}
If $\norm{\cdot}$ is a norm on a real or complex vector space $X$, then
$$
d(x,y) = \norm{x-y} \qquad \forall x,y \in X
$$
defines a metric $d$ on $X$.
{% endproposition %}
{% definition Norm Topology %}
If $(X,\norm{\cdot})$ is a normed space,
then the topology on $X$ associated with the metric induced by the norm
is called the *norm topology*.
{% enddefinition %}
TODO equivalent norms, subspaces, product, quotient
TODO open ball, closed ball, sphere
{% proposition Continuity of Addition, Scalar Multiplication %}
Let $(X,\norm{\cdot})$ 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, $\KK \times X \to X$, $(\alpha,x) \mapsto \alpha x$,
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.
{% endproposition %}
{% proof %}
This follows from the inequalities
$$
\norm{(x+y) - (x_0+y_0)} \le \norm{x-x_0} + \norm{y-y_0}
$$
and
$$
\norm{\alpha x - \alpha_0 x_0} \le \abs{\alpha} \norm{x-x_0} + \abs{\alpha-\alpha_0} \norm{x_0}.
$$
fix
{% endproof %}
{% proposition Uniform Continuity of the Norm %}
Let $(X,\norm{\cdot})$ be a normed space.
Then we have
$$
\big\lvert \norm{x} - \norm{y} \big\rvert \le \norm{x-y} \qquad \forall x,y \in X.
$$
Consequently, the norm is a uniformly continous function with respect to the norm topology.
{% endproposition %}
{% proposition Quotient Norm %}
Let $Y$ be a closed linear subspace of a normed space $(X,\norm{\cdot})$.
Then
$$
\norm{x+Y} = \inf_{y \in Y} \norm{x+y} \qquad \forall x \in X
$$
defines a norm on the quotient vector space $X/Y$.
{% endproposition %}
## Duals of Normed Spaces
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