Asked Feb 22, 2019

Recall that we say that a random variable X is in the vector space L^2 if it has finite second moment, EX^2 In this problem we will understand a bit better the geometry of the vector space L^2
(1) Show that ||X||_2 = √EX^2 is a norm


Expert Answer

Step 1

Properties of a norm:

In order to prove that ||X||2 = √E(X2) is a norm, it must satisfy 3 conditions:

  • Positivity,
  • Homogeneity,
  • Sub-additivity.
Step 2

Proof of positivity:

To prove: ||x||2 ≥ 0 for all x ϵ X and x = 0 if and only if x = 0.


Step 3

Proof of homogeneity:

To prove: For all x ϵ X, ||cx||2 = |c|.||x||2 for all real c...


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