Functional Analysis

By the end of this course, students should be able to: 

• describe properties of normed linear spaces and construct examples of such spaces
• extend basic notions from calculus to metric spaces and normed vector spaces
• state and prove theorems about finite dimensionality in normed vector spaces
• state and prove the Cauchy-Swartz Inequality and apply it to the derivation of other inequalities
• distinguish pointwise and uniform convergence
• prove that a given space is a Hilbert spaces or a  Banach Spaces
• describe the dual of a normed linear space
• apply orthonormality to Fourier series expansions of functions
• state and prove the Hahn-Banach theorem