Hilbert space Study Guide

📖 Core Concepts Hilbert space – a real or complex inner‑product space that is complete (every Cauchy sequence converges). Inner product ⟨x,y⟩ – conjugate‑symmetric, linear in the first argument, positive‑definite; it defines the norm ‖x‖ = √⟨x,x⟩ and distance d(x,y)=‖x−y‖. Completeness – guarantees limits of Cauchy sequences (or absolutely convergent series ∑‖xₙ‖) stay inside the space. Orthonormal basis – a set {eₖ} with ⟨eₖ,eⱼ⟩=δₖⱼ, ‖eₖ‖=1, whose linear span is dense; every element has a unique Fourier expansion. Projection theorem – every closed subspace V has a unique orthogonal projection PV; x = Pv + (x−Pv) with x−Pv ⟂ V. Riesz representation – each continuous linear functional φ on H is φ(x)=⟨x,u⟩ for a unique u∈H. Self‑adjoint operator – A = A; its spectrum lies in ℝ, and in the bounded case it admits a spectral resolution T = ∫ λ dEλ. Unitary operator – UU = UU = I; preserves norms and inner products (‖Ux‖=‖x‖). 📌 Must Remember Cauchy–Schwarz: |⟨x,y⟩| ≤ ‖x‖‖y‖. Parallelogram law: ‖x+y‖² + ‖x−y‖² = 2‖x‖² + 2‖y‖². Polarization (real): ⟨x,y⟩ = ¼(‖x+y‖² − ‖x−y‖²). Polarization (complex): ⟨x,y⟩ = ¼(‖x+y‖² − ‖x−y‖² + i‖x+iy‖² − i‖x−iy‖²). Bessel’s inequality: ∑{k=1}^n |⟨x,fk⟩|² ≤ ‖x‖² for any orthonormal set {fk}. Parseval’s identity: ‖x‖² = ∑{k∈B} |⟨x,ek⟩|² for an orthonormal basis {ek}. Operator norm: ‖A‖ = sup{‖x‖=1} ‖Ax‖. Adjoint definition: ⟨Ax,y⟩ = ⟨x,A y⟩ ∀x,y. Self‑adjoint ⇒ spectrum ⊂ ℝ; for compact self‑adjoint operators eigenvalues form a countable set accumulating only at 0. 🔄 Key Processes Finding the orthogonal projection onto a closed subspace V Solve ⟨x−Pv, v'⟩ = 0 ∀v'∈V. In a basis {ek} of V, Pv = Σ ⟨x,ek⟩ ek. Fourier expansion of a vector Compute coefficients ck = ⟨x, ek⟩. Reconstruct: x = Σ ck ek (converges in ‖·‖). Spectral decomposition of a bounded self‑adjoint operator T Build the spectral family {Eλ}. Evaluate T = ∫ λ dEλ (Stieltjes integral). Applying the Riesz representation Given φ∈H, find u such that φ(x)=⟨x,u⟩ (solve linear equations in orthonormal basis). Checking boundedness of an operator Verify existence of C with ‖Ax‖ ≤ C‖x‖ ∀x (equivalently, compute ‖A‖). 🔍 Key Comparisons Bounded vs. Unbounded operator – Bounded ⇔ continuous on whole H; unbounded operators are defined only on a dense domain D(T) and may not be continuous. Self‑adjoint vs. Normal – Self‑adjoint: A = A (real spectrum). Normal: AA = AA (includes unitary, self‑adjoint, and others). Compact vs. General bounded operator – Compact maps bounded sets to relatively compact sets; spectrum of compact self‑adjoint operator consists of eigenvalues →0, whereas a general bounded operator may have continuous spectrum. ℓ² vs. L² – ℓ²: square‑summable sequences; L²: square‑integrable functions w.r.t. a measure. Both are Hilbert spaces with analogous inner products. ⚠️ Common Misunderstandings “Every Hilbert space is finite‑dimensional.” – False; many (e.g., L², ℓ²) are infinite‑dimensional but still complete. “Orthogonal projection always exists for any subset.” – It exists only for closed linear subspaces. “Self‑adjoint ⇒ bounded.” – Only true for operators defined on the whole space; many important self‑adjoint operators (e.g., momentum) are unbounded. “Parseval’s identity works for any orthonormal set.” – It requires a complete orthonormal basis; otherwise only Bessel’s inequality holds. 🧠 Mental Models / Intuition Hilbert space ≈ “infinite‑dimensional Euclidean space.” Distances, angles, orthogonal projections behave just like in ℝⁿ. Projection theorem = “dropping a perpendicular.” The best approximation to x from V is the foot of the perpendicular. Riesz representation = “inner‑product fingerprint.” Every continuous linear functional is just “inner product with a fixed vector.” Spectrum of a self‑adjoint operator = “generalized eigenvalues.” Think of them as the possible measurement outcomes in quantum mechanics. 🚩 Exceptions & Edge Cases Non‑separable Hilbert spaces – May lack a countable orthonormal basis; most physics applications assume separability. Zero eigenvalue for compact self‑adjoint operator – Always present (limit point of eigenvalues). Unbounded operators – Domain D(T) must be dense; self‑adjointness requires D(T)=D(T) and equality of graphs. Weak convergence – Does not imply norm convergence, but every weakly convergent sequence is bounded. 📍 When to Use Which Least‑squares / best approximation → use orthogonal projection onto a closed subspace (Hilbert projection theorem). Expanding a function → pick an orthonormal basis suited to the problem (e.g., Fourier basis for periodic functions, eigenfunctions of a Sturm–Liouville problem). Studying spectra → apply spectral theorem for bounded self‑adjoint operators; for compact operators, focus on eigenvalue sequences. Representing linear functionals → invoke Riesz representation instead of solving a system of equations. Quantum observables → model with (possibly unbounded) self‑adjoint operators; use spectral decomposition to compute measurement probabilities. 👀 Patterns to Recognize Orthogonal series convergence – If Σ‖uk‖² converges, then Σuk converges (Pythagorean series test). Norm identities – Whenever you see ‖x±y‖², think of parallelogram or polarization formulas. Projection idempotence – Any operator P with P²=P and P=P is an orthogonal projection onto its range. Compact + self‑adjoint – Expect a discrete real spectrum accumulating only at 0. Tensor product inner product – Always factorizes: ⟨x₁⊗x₂, y₁⊗y₂⟩ = ⟨x₁,y₁⟩⟨x₂,y₂⟩. 🗂️ Exam Traps Confusing Bessel’s inequality with Parseval’s identity – Bessel’s holds for any orthonormal set; Parseval requires a complete basis. Assuming every self‑adjoint operator is bounded – In quantum mechanics many observables are unbounded; the domain matters. Misreading “unitary” as “self‑adjoint”. Unitary operators preserve inner products but are not necessarily equal to their adjoint. Taking “closed subspace” for “any subspace”. Orthogonal projections exist only for closed subspaces; an open subspace may not admit a projection. Forgetting the conjugate symmetry in complex inner products – ⟨x,y⟩ = overline{⟨y,x⟩}; dropping the conjugate leads to sign errors in calculations. --- This guide condenses the highest‑yield material from the outline. Review each bullet, practice the key processes, and watch for the listed traps to boost confidence on the exam.