Functional Analysis — Selected Topics

Banach and Hilbert Spaces

A Banach space is a complete normed vector space. A Hilbert space is a Banach space equipped with an inner product inducing the norm. Key examples: L²(Ω), Sobolev spaces H^k(Ω), and sequence spaces ℓ².

Bounded Linear Operators

A linear operator T: X → Y between normed spaces is bounded if ‖T‖ = sup{‖Tx‖ : ‖x‖ ≤ 1} < ∞. Key results include the Uniform Boundedness Principle, Open Mapping Theorem, and Closed Graph Theorem.

Spectral Theory

For a bounded operator T on a Banach space, the spectrum σ(T) consists of all λ for which (T − λI) is not invertible. The spectral radius r(T) = sup{|λ| : λ ∈ σ(T)} satisfies r(T) = lim ‖Tⁿ‖^1/n.

For self-adjoint operators on Hilbert spaces, the spectrum is real and the spectral theorem provides a diagonalization via projection-valued measures.

Applications to PDEs

Functional analytic tools underpin the modern theory of PDEs: Lax–Milgram for existence of weak solutions, Fredholm theory for elliptic operators, and semigroup theory for parabolic evolution equations.