Analysis 3 Proofs and Definitions Weeks 3 and 4

Why continuity is crucial for fubini’s theorem? Give an example

C1 is not preserved under uniform convergence (once differentiable functions on an interval)

Prove

Convergence and uniform convergence of series of functions

Prove integrate term by term theorem

Prove differentiate term by term theorem

What is the Weierstrass M test? prove it

Give an example of a function that is continuous but not differentiable anywhere

What is the euclidean norm (magnitude)?

Prove Cauchy Schwarz Inequality

Prove the triangle inequality

Define convergence in vector space

Prove the uniqueness of a limit in vector space

What is component wise convergence? Prove it

What is the max norm and Manhattan norm?

Prove the equivalence of norms in terms of convergence

If xj converges to x then it also converges in the euclidean norm also show that converse isn’t true with the help of an example

All convergent sequences are bounded

All cauchy sequences are convergent