Definition:Fréchet Space (Functional Analysis)
(Redirected from Definition:Fréchet Product Metric)
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This page is about Fréchet Space in the context of Functional Analysis. For other uses, see Fréchet Space.
Definition
Let $\R^\omega$ denote the countable-dimensional real Cartesian space.
Let:
- $x := \family {x_i}_{i \mathop \in \N} = \tuple {x_0, x_1, x_2, \ldots}$
and:
- $y := \family {y_i}_{i \mathop \in \N} = \tuple {y_0, y_1, y_2, \ldots}$
denote arbitrary elements of $\R^\omega$.
Let the distance function $d: \R^\omega \times \R^\omega \to \R$ be applied to $\R^\omega$ as:
- $\forall x, y \in \R^\omega: \map d {x, y} = \ds \sum_{i \mathop \in \N} \dfrac {2^{-i} \size {x_i - y_i} } {1 + \size {x_i - y_i} }$
The distance function $d$ is referred to as the Fréchet (product) metric.
The resulting metric space $\struct {\R^\omega, d}$ is then referred to as the Fréchet (metric) space.
Also see
- Fréchet Space (Functional Analysis) is Metric Space, which demonstrates that $d$ is indeed a metric on $\R^\omega$
- Results about the Fréchet product metric can be found here.
- Not to be confused with Definition:Fréchet Space (Topology).
Source of Name
This entry was named for Maurice René Fréchet.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $37$. Fréchet Space: $7$