Fréchet metric

Fréchet metric (after Maurice René Fréchet ) is a term from functional analysis . It creates a connection between metric and norm .

definition

Let be any real or complex vector space . A Fréchet metric is a function that satisfies for the following conditions: ${\displaystyle V}$ ${\displaystyle \varrho \colon V\to \mathbb {R} }$ ${\displaystyle x,y\in V}$

1. ${\displaystyle \varrho (x)=\varrho (-x)}$
2. ${\displaystyle \varrho (x)\geq 0}$, in which ${\displaystyle \varrho (x)=0\iff x=0}$
3. ${\displaystyle \varrho (x+y)\leq \varrho (x)+\varrho (y)}$

That is, is symmetric, nonnegative, and satisfies the triangle inequality . ${\displaystyle \varrho }$

Examples

• Every norm on is a Fréchet metric, because obviously fulfills the conditions (2) and (3). The validity of (1) follows from the homogeneity of standards. However, the converse is not true: for example, the Fréchet metric is not a norm because it is not homogeneous. ${\displaystyle x\mapsto \|x\|}$ ${\displaystyle V}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle V=\mathbb {R} ^{n}}$
  ${\displaystyle \varrho (x):={\frac {|x|}{1+|x|}}}$
  
• If there is a countable family of semi-norms on the vector space with the property for all then it is defined by a Fréchet metric which generates the same topology as the family of semi-norms. ${\displaystyle (p_{k})_{k\in \mathbb {N} }}$ ${\displaystyle V}$
  ${\displaystyle p_{k}(x)=0\,}$ ${\displaystyle k\in \mathbb {N} \Longrightarrow x=0,}$
  
  ${\displaystyle \varrho (x)=\sum _{k\in \mathbb {N} }{\frac {1}{2^{k}}}{\frac {p_{k}(x)}{1+p_{k}(x)}}}$
  
• The spaces for equipped with the Fréchet metric are examples of generally non- locally convex spaces . ${\displaystyle L^{p}}$ ${\displaystyle 0<p<1}$
  ${\displaystyle \varrho _{p}(f):=\int _{\Omega }\left\|f(x)\right\|^{p}\,\mathrm {d} \mu (x)}$
  

Applications

• A metric can be defined in a vector space by means of a Fréchet metric . The fact that the mapping defined in this way is a metric follows directly from the definition of the Fréchet metric. ${\displaystyle d(x,y):=\varrho (x-y)}$
• The reverse applies: Every metric on a vector space that is translation invariant, i.e. H. , is created by precisely such a Fréchet metric. ${\displaystyle d}$ ${\displaystyle d(x+c,y+c)=d(x,y)}$
• A ( Hausdorff's ) topological vector space has a Fréchet metric that generates its topology when it is first countable .
• If a (real or complex) vector space with Fréchet metric has the additional properties that it is complete and that the topology of this vector space is locally convex , then it is a Fréchet space .

literature

• HW Alt: Linear functional analysis. 4th edition, Springer, Berlin 2002, ISBN 3-540-43947-1 .

Individual evidence

1. ↑ H. W. Alt: Lineare functional analysis. An application-oriented introduction . 6th edition. Springer-Verlag, Berlin, Heidelberg 2012, ISBN 978-3-642-22260-3 , Chapter 2. Subsets of function spaces, U2.11, p. 140 .