MU-304 Functional Analysis

Unit · year 3

MU-304 · Functional Analysis

Threads space · structure24 lectures5 theorems

Infinite-dimensional linear algebra and analysis on Banach and Hilbert spaces.

PREREQUISITES

L01	Normed Vector Spaces —
L02	Banach Spaces and Completeness —
L03	Examples: ℓᵖ, Lᵖ, C[a,b] —
L04	Bounded Linear Operators —
L05	The Operator Norm and B(X,Y) —
L06	Finite-Dimensional Spaces and Equivalence of Norms —
L07	Dual Spaces The Hahn–Banach theorem
L08	The Hahn–Banach Theorem The Hahn–Banach theorem
L09	Consequences of Hahn–Banach The Hahn–Banach theorem
L10	Weak and Weak* Topologies —
L11	The Banach–Alaoglu Theorem —
L12	Baire Category in Banach Spaces The uniform boundedness principle
L13	The Uniform Boundedness Principle The uniform boundedness principle
L14	Applications: Divergence of Fourier Series The uniform boundedness principle
L15	The Open Mapping Theorem The open mapping theorem
L16	The Bounded Inverse Theorem The open mapping theorem
L17	The Closed Graph Theorem The closed graph theorem
L18	Hilbert Spaces and Orthonormal Bases —
L19	The Projection Theorem in Hilbert Space The Riesz representation theorem
L20	The Riesz Representation Theorem The Riesz representation theorem
L21	Adjoints and Self-Adjoint Operators —
L22	Compact Operators —
L23	The Spectral Theorem for Compact Self-Adjoint Operators —
L24	Synthesis: Completeness Turned into Theorems The open mapping theorem, The uniform boundedness principle

Bounded functionals extend without increasing norm.

A surjective bounded operator between Banach spaces is open.

A closed-graph operator between Banach spaces is bounded.

Pointwise-bounded families of operators are uniformly bounded.

Every bounded functional on a Hilbert space is an inner product.