Hilbert space: treating functions as vectors
10 days ago
- #quantum mechanics
- #functional analysis
- #linear algebra
- Hilbert space allows applying linear algebra tools to functions by treating them as infinite-dimensional vectors.
- Functions can be viewed as vectors with infinite dimensions, where each index corresponds to a real number.
- The set of all functions mapping a set to real or complex numbers forms a vector space with standard addition and scalar multiplication.
- Square integrable functions, denoted as L², form a subspace of this vector space, crucial for defining Hilbert space.
- An inner product for functions in L² is defined, generalizing the concept from finite-dimensional vectors to functions.
- The inner product space of square integrable functions is complete, making it a Hilbert space, essential for convergence properties.
- Hilbert spaces enable generalized Fourier series, allowing functions to be expressed as sums of orthogonal basis functions.
- Quantum mechanics utilizes Hilbert spaces to describe particle states, with inner products representing probabilities.
- Proofs for vector space axioms and subspace properties of square integrable functions are provided in appendices.