Index
Smooth and distribution vectors: a reading path
From continuous unitary actions to smooth domains, generators, distribution vectors, and the Heisenberg example.
Core idea
Follow the passage from a unitary group action on a Hilbert space to its infinitesimal action on smooth vectors. The common domain and its topology are part of the construction.
Continuity and the hypotheses
- Strongly continuous unitary representation
- Why weak and strong continuity agree here
- Irreducibility uses closed invariant subspaces
- A noncompact, nonabelian type I group with a two-dimensional irreducible representation
- Strong continuity without operator-norm continuity
Smoothness and the infinitesimal action
- Finite differentiability of a vector
- Smooth vector
- Gårding smoothing subspace and density theorem
- Derived Lie-algebra representation
- The explicit Fréchet topology and derivative estimates
- Smooth vectors as the intersection of iterated generator domains
- Stone's theorem, generators, and essential skew-adjointness
- Why a core determines a closed operator
Generalized vectors and nuclearity
- Continuous anti-dual
- Distribution vectors and the extended action
- Rigged Hilbert space and its nuclear convention
- Trace-class representation
- The nuclearity criterion and why type I is insufficient
The concrete Heisenberg model
- Heisenberg group and Schrödinger representation
- Why its smooth vectors are precisely Schwartz functions
- Derivative, multiplication, and the central commutator
- Fundamental fields for a left action
- When the derived action is a Lie derivative
The stronger operator-norm framework
For dynamical applications, continue to the Koopman and spectrum reading path. The number-field translation reading path covers the separate arithmetic example, including single translations and joint actions.