proving $L^p$ as a normed vector space for $p > 1$.

Asked Sep 09 '21 at 19:47

Active Sep 09 '21 at 19:47

Viewed 74 times

I know that to prove a vector space is a normed vector space we need to prove the homogeneity, point separation and sub-additivity of the vector space. I am able to prove it for $p = 1$ (a trivial case by taking mean) and for $p = \infty$ (by taking $\lVert X \rVert_ \infty = sup\{ |X(\omega)|: \omega \in \Omega\}$).

While trying to prove it for $p>1$ I tried following:

As $\lVert aX \rVert = |a||X|$, so $(E|aX|^p)^{1/p} = (|a|^p.E|X|^p)^{1/p} = |a| \lVert X \rVert _p$.
$\lVert X \rVert_p \ge 0$ because $E|X|^p \ge 0$ and hence $(E|aX|^p)^{1/p} \ge 0$.

But I am finding it difficult to prove the sub-additivity or triangular inequality. Can anyone give a hint or source to read more about it? Thanks in advance!

asked Sep 09 '21 at 19:47

indmath

You want to google the Minkowski inequality, which is precisely the triangle inequality for L^p spaces – peek-a-boo Sep 09 '21 at 19:49
Does this answer your question? How to prove triangle inequality for $p$-norm? (Note that the or triangular inequality is proved in one of the answers) – Arctic Char Sep 09 '21 at 19:57

proving $L^p$ as a normed vector space for $p > 1$.

0 Answers0