Mwaka wa kwanza wa uzamiliAnalysisImepangwa
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
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 42 (hadi chini kabisa)
Utajifunza
- 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.
Fasili
- 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.
Nadharia
- 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.
Mawazo potofu ya kawaida
- Assuming every linear map between normed spaces is continuous.
- Thinking all norms on an infinite-dimensional vector space are equivalent.
Inaelekea kwa
Ugunduzi wa hiari
Njia za kujifunza
Usomaji zaidi
- 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)