Hilbert Spaces

A Hilbert space is a complete inner-product space — a Banach space whose norm comes from an inner product, so that geometry (angles, orthogonality, projection) is available in infinite dimensions. It is the state space of quantum mechanics: the inner product gives probability amplitudes, orthonormal bases give measurement outcomes, and the projection theorem gives the collapse rule. This page builds on normed-banach.md and grounds QM/preliminaries.md.

Inner products

An inner product on a complex vector space $\mathcal{H}$ is a map $⟨\cdot ,\cdot ⟩:\mathcal{H}\times \mathcal{H}\to \mathbb{C}$ that is linear in one argument, conjugate-symmetric $⟨x,y⟩=\overline{⟨y,x⟩}$, and positive-definite $⟨x,x⟩>0$ for $x\ne 0$. It induces the norm $\Vert x\Vert =\sqrt{⟨x,x⟩}$ and obeys the Cauchy–Schwarz inequality $|⟨x,y⟩|\le \Vert x\Vert \Vert y\Vert$.

Physics convention. Following QM/preliminaries.md, the inner product is linear in the second argument (antilinear in the first), matching Dirac's $⟨\varphi |\psi ⟩$. Mathematicians usually take the first argument linear; only the placement of the conjugate differs.

A Hilbert space is an inner-product space that is complete in the induced norm. The parallelogram law $\Vert x+y{\Vert }^2+\Vert x-y{\Vert }^2=2\Vert x{\Vert }^2+2\Vert y{\Vert }^2$ characterizes exactly which Banach norms come from an inner product.

Orthogonality and bases

The inner product gives orthogonality ($x\perp y$ iff $⟨x,y⟩=0$) and the Pythagorean theorem. An orthonormal basis $\{e_n\}$ is a maximal orthonormal set; then every vector expands as $$x=\sum _n⟨e_n,x⟩e_n,\Vert x{\Vert }^2=\sum _n|⟨e_n,x⟩{|}^2(\text{Parseval}).$$ A Hilbert space is separable (has a countable orthonormal basis) in essentially all cases physics uses; then it is isomorphic to ${\ell }^2$. Fourier series are exactly the orthonormal expansion in $L^2$ of the circle — the same Peter–Weyl content noted in group-theory/compact-groups.md.

The projection theorem

The defining geometric feature:

Projection theorem. For a closed subspace $M\subseteq \mathcal{H}$, every $x$ decomposes uniquely as $x=m+m^{\perp }$ with $m\in M$ and $m^{\perp }\in M^{\perp }$. The map $P_M:x\mapsto m$ is the orthogonal projection, the unique closest point of $M$ to $x$.

Orthogonal projections $P=P^2=P^{†}$ are the mathematical form of ideal measurements: projecting a state onto an eigenspace is the QM collapse (physics-bridge.md). The decomposition $\mathcal{H}=M\oplus M^{\perp }$ has no analogue in a general Banach space — it is what the inner product buys.

The Riesz representation theorem

Hilbert spaces are self-dual:

Riesz representation theorem. Every bounded linear functional $\phi \in >{\mathcal{H}}^{\ast }$ is $\phi (x)=⟨y,x⟩$ for a unique $y\in >\mathcal{H}$; the map $\phi \leftrightarrow y$ is a conjugate-linear isometry ${\mathcal{H}}^{\ast }\cong \mathcal{H}$.

This is the rigorous meaning of the bra–ket correspondence: every bra $⟨\varphi |$ (a functional) is an inner product with a unique ket $|\varphi ⟩$ (a vector). Dirac notation is Riesz representation made typographical. (For the improper kets $|x⟩$, $|p⟩$ — which are not in $\mathcal{H}$ — one needs the rigged-space extension of rigged-hilbert-space.md.)

Tensor products

The composite of two quantum systems uses the Hilbert-space tensor product ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$: the completion of the algebraic tensor product in the inner product $⟨a\otimes b,c\otimes d⟩=⟨a,c⟩⟨b,d⟩$. Non-product vectors are entangled states. This is the structure behind identical-particle Fock space (QFT/fock-space-inventory.md) and the QM tensor-product postulate.

References

• Reed & Simon, Methods of Modern Mathematical Physics I, Ch. II.
• Rudin, Functional Analysis, Ch. 12.
• von Neumann, Mathematical Foundations of Quantum Mechanics — the original.