研究生一年级Analysis计划中

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

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前置知识

Measure theory and integration Advanced linear algebra

42 项必需的前置知识（一直追溯到底）

你将学会

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.

定义

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.

定理

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.

常见误解

Assuming every linear map between normed spaces is continuous.
Thinking all norms on an infinite-dimensional vector space are equivalent.

通向

Mathematical foundations of quantum mechanics

可选发现

学习路径

Graduate preparation