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Triangle inequality for the Lp norm.
Minkowski inequality: Let be a measure space and let . For all ,
In particular, when this reads
This is the triangle inequality for the Lp norm and is what makes Lp spaces into normed linear spaces for . For it is the triangle inequality for the essential supremum norm on essentially bounded functions.
A norm on a measure space (for ) assigns to a measurable function the value
whenever the Lebesgue integral of the nonnegative function is finite; here uses the absolute value. For one defines
using the essential supremum.
If and are equal almost everywhere, then , so the norm is naturally a norm on the corresponding space.
For a measure space and , the space is
where is or , is the norm, and means that almost everywhere.
The quotient makes the norm a genuine norm rather than a seminorm. The cases and correspond to functions and functions, respectively.
The essential supremum of a measurable function on a measure space is the extended real number
A function on a measure space is a measurable function or such that
Here denotes the essential supremum.
If and are equal almost everywhere, then , so membership in depends only on the equivalence class modulo a null set. These equivalence classes form the case of an space.