Definition:Fréchet Space (Functional Analysis)

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:
 * $\forall x, y \in \R^\omega: \map d {x, y} = \displaystyle \sum_{i \mathop \in \N} \dfrac {2^{-i} \size {x_i - y_i} } {1 + \size {x_i - y_i} }$

The distance function $d$ is referred to as the Fréchet (product) metric.

The resulting metric space $\styruct {\R^\omega, d}$ 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 $\R^\omega$


 * Definition:Hilbert Sequence Space