Category:Definitions/P-Product Metrics

From ProofWiki
Jump to navigation Jump to search

This category contains definitions related to $p$-product metrics.
Related results can be found in Category:P-Product Metrics.


Let $M_{1'} = \struct {A_{1'}, d_{1'} }$ and $M_{2'} = \struct {A_{2'}, d_{2'} }$ be metric spaces.

Let $A_{1'} \times A_{2'}$ be the cartesian product of $A_{1'}$ and $A_{2'}$.

Let $p \in \R_{\ge 1}$.


The $p$-product metric on $A_{1'} \times A_{2'}$ is defined as:

$\map {d_p} {x, y} := \paren {\paren {\map {d_{1'} } {x_1, y_1} }^p + \paren {\map {d_{2'} } {x_2, y_2} }^p}^{1/p}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_{1'} \times A_{2'}$.


The metric space $\mathcal M_p := \struct {A_{1'} \times A_{2'}, d_p}$ is the $p$-product (space) of $M_{1'}$ and $M_{2'}$.

Subcategories

This category has only the following subcategory.