when the integral exists. The set of functions with finite -norm forms a vector space with the usual pointwise addition and scalar multiplication of functions. In particular, the set of functions with zero -norm form a linear subspace of , which for this article will be called . We are then interested in the quotient space , which consists of complex functions on with finite -norm, identified up to equivalence almost everywhere. This quotient space is the complex -space on .
If , the vector space is complete with respect to the norm.
The space .
The space is somewhat special, and may be defined without explicit reference to an integral. First, the -norm of is defined to be the essential supremum of :
However, if is the trivial measure, then essential supremum of every measurable function is defined to be 0.
The definitions of , , and then proceed as above, and again we have that is complete. Functions in are also called essentially bounded.
Let and . Then but .
|Date of creation||2013-03-22 12:21:32|
|Last modified on||2013-03-22 12:21:32|
|Last modified by||Mathprof (13753)|
|Synonym||essentially bounded function|