\begin{definition}[Definition:Fréchet Space (Functional Analysis)]
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'''.
\end{definition}