The Lp Spaces

Let be a measure space and let . We write for the set of all measurable functions such that is integrable. For such we write

Endowed with pointwise addition and scalar multiplication becomes a vector space.

Proof   We show that is a linear subspace of the vector space of all -valued functions on . The set is nonempty since it contains the zero function. Now, suppose and are in . Then the sum is measurable, because and are measurable. Moreover, the function is integrable, because we have the estimate

where and are integrable. This proves that lies in . Finally, it is easy to see that lies in for any scalar .

is a seminorm on .

Young Inequality

Consider such that . Then