स्नातकोत्तर प्रथम वर्षAnalysisपाठ उपलब्ध
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
इस पर आधारित
42 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- 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.
परिभाषा
- 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.
प्रमेय
- 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.
आम भ्रांतियाँ
- Assuming every linear map between normed spaces is continuous.
- Thinking all norms on an infinite-dimensional vector space are equivalent.
आगे ले जाता है
वैकल्पिक खोजें
अधिगम पथ
आगे पढ़ने के लिए
- 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)