Analysis 3 Proofs and Definitions Weeks 7 and 8

What does it mean to be sequentially compact

Prove that the 2 definitions of sequential compactness are equivalent

Continuous function applied on a compact set is also compact

Extreme value theorem

Corollary of extreme value theorem on k dimensional vector about bounds

Linear map notation for clarity

Frobenius norm and operator norm, and how they relate to each other

Comparison of Frobenius and Operator norm

Prove that operator norm is a norm

What is the composition bound?

Proving injectivity of a set using alpha and bounds of norms

Convergence and continuity in linear maps

What is a directional derivative?

Existence of directional derivative does not imply continuity along lines

Frechet derivative for linear approximations

What does differentiability even mean in higher dimensions?

A from differentiability is unique if it exists

A is the frechet derivative of f at x

If f is Frechet differentiable at p then it is continuous at p

What is component wise differentiability?

If the frechet derivative exists then the linear derivatives exist but the converse isn’t true

Partial derivative of a vector

Jacobian matrix or gradient matrix

How is the frechet derivative related to the jacobian?

Different notations for the same thing and the different derivatives are analogous for different continuities