Hilbert Space
A complete abstract vector space possessing an inner product that allows length and angle to be defined for infinite-dimensional function spaces.
Governing formula
⟨f, g⟩ = ∫ f*(x) g(x) dx, ||f|| = √(⟨f, f⟩)
SI unit
Complete Inner Product Space
In depth
Serves as the rigorous mathematical state space for quantum mechanics. Quantum wave functions represent state vectors in Hilbert space, and physical observables are self-adjoint Hermitian operators.
Examples in the real world
Expanding quantum wave functions in complete orthonormal bases in L² Hilbert function space.