We continue our project to show that the spaces are actually Banach spaces with Minkowski’s inequality. This will allow us to conclude that is a normed vector space. It states that if and are both in , then their sum is in , and we have the inequality
We start by considering Hölder’s inequality in a toy space I’ll whip up right now. Take two isolated points, and let each one have measure ; the whole space of both points has measure . A function is just an assignment of a pair of real values , and integration just means adding them together. Hölder’s inequality for this space tells us that
where and are Hölder-conjugate to each other. We can set , , and and use this inequality to find
Dividing out and raising both sides to the th power, we conclude that . Thus if both and are integrable, then so is . Thus must be in .
Now we calculate
Dividing out by we find that
This lets us conclude that is a vector space. But we can also verify the triangle identity now. Indeed, if , , and are all in , then Minkowski’s inequality shows us that
which is exactly the triangle inequality we want. Thus is a norm, and is a normed vector space.