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The difference of norms is bounded by the norm of the difference
Reverse triangle inequality: In a normed vector space , for all ,
Equivalently,
This inequality is frequently used to show continuity of the norm map and to transfer convergence of vectors to convergence of their norms.
A normed vector space is a vector space over or , equipped with a norm on .
Let be a function between metric spaces, and let . We say is continuous on if the restriction , with carrying the subspace metric, is continuous. Equivalently, is continuous at every point .
Spelled out: for every and every , there exists such that for all ,
A sequence in a topological space converges to if, for every neighborhood of , there is such that for all .
The triangle inequality states that for any three points in a metric space :
This is one of the defining axioms of a metric.