First-year graduateAnalysisPlanned
Functional analysis
Normed, Banach and Hilbert spaces, bounded operators, the Hahn–Banach, open mapping, closed graph and uniform boundedness theorems, weak topologies, and the spectral theorem for compact self-adjoint operators.
math.analysis.functional-analysis
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 42
You will learn to
- Prove the Riesz representation theorem for Hilbert spaces and work with orthonormal bases.
- State and apply the Hahn–Banach, open mapping and uniform boundedness theorems.
- Prove the spectral theorem for compact self-adjoint operators.
Definition
- A Banach space is a complete normed vector space; a Hilbert space is a complete inner product space.
- A linear operator T between normed spaces is bounded if ‖Tx‖ ≤ C‖x‖ for some constant C and all x.
- A bounded operator is compact if it maps bounded sets to sets with compact closure.
Theorem
- Riesz: every bounded linear functional on a Hilbert space H has the form x ↦ ⟨x, y⟩ for a unique y ∈ H.
- Uniform boundedness principle.
- Spectral theorem: a compact self-adjoint operator on a Hilbert space has an orthonormal basis of eigenvectors for the closure of its range, and its non-zero eigenvalues can only accumulate at 0.
- The closed unit ball of an infinite-dimensional normed space is not compact in the norm topology.
Common misconceptions
- Assuming every linear map between normed spaces is continuous.
- Thinking all norms on an infinite-dimensional vector space are equivalent.
Leads to
Optional discoveries
Learning pathways
Further reading
- Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011)
- Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis (1972)