2.1 Definition and properties of Hilbert spaces

Hilbert spaces form the foundation of spectral theory, extending Euclidean space to infinite dimensions. These spaces combine algebraic and topological structures, enabling the study of complex mathematical and physical systems.

Key properties of Hilbert spaces include completeness, separability, and orthogonality. These features allow for powerful techniques in analyzing linear operators, forming the basis for advanced concepts in functional analysis and spectral theory.

Definition of Hilbert spaces

• Hilbert spaces form the foundation of spectral theory providing a framework for analyzing linear operators
• These spaces extend the concept of Euclidean space to infinite dimensions enabling the study of complex mathematical and physical systems
• Hilbert spaces combine algebraic and topological structures crucial for understanding spectral properties of operators

Inner product spaces

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• Vector spaces equipped with an inner product function $\langle x, y \rangle$ mapping pairs of vectors to scalars
• Inner product satisfies conjugate symmetry $\langle x, y \rangle = \overline{\langle y, x \rangle}$
• Linearity in the first argument $\langle ax + by, z \rangle = a\langle x, z \rangle + b\langle y, z \rangle$
• Positive definiteness $\langle x, x \rangle \geq 0$ with equality if and only if $x = 0$
• Induces a norm $\|x\| = \sqrt{\langle x, x \rangle}$ and metric $d(x,y) = \|x-y\|$

Completeness property

• Every Cauchy sequence in a Hilbert space converges to a limit within the space
• Ensures the existence of limits for certain infinite series and improves analytical tools
• Distinguishes Hilbert spaces from general inner product spaces (pre-Hilbert spaces)
• Allows the application of powerful theorems like the Riesz representation theorem

Separability in Hilbert spaces

• Contains a countable dense subset enabling representation by countable bases
• Separable Hilbert spaces possess orthonormal bases (Hilbert bases)
• Most commonly encountered Hilbert spaces in applications are separable ($L^2$ spaces)
• Simplifies many theoretical results and practical computations in spectral theory

Key properties of Hilbert spaces

• Hilbert spaces combine algebraic structure (vector space) with topological properties (completeness)
• These properties enable powerful techniques for analyzing linear operators central to spectral theory
• Understanding these key properties forms the basis for more advanced concepts in functional analysis

Orthogonality and orthonormality

• Two vectors $x$ and $y$ are orthogonal if their inner product equals zero $\langle x, y \rangle = 0$
• Orthonormal sets consist of mutually orthogonal unit vectors
• Gram-Schmidt process constructs orthonormal bases from linearly independent sets
• Orthogonal decomposition theorem states any vector can be uniquely expressed as a sum of its projection onto a closed subspace and its orthogonal complement

Projection theorem

• For any closed subspace $M$ of a Hilbert space $H$ and any vector $x \in H$, there exists a unique vector $y \in M$ such that $\|x - y\| \leq \|x - z\|$ for all $z \in M$
• The vector $y$ is called the orthogonal projection of $x$ onto $M$
• Projection operator $P_M: H \to M$ is linear and satisfies $P_M^2 = P_M$ (idempotent)
• Fundamental for approximation theory and solving operator equations

Riesz representation theorem

• For any continuous linear functional $f$ on a Hilbert space $H$, there exists a unique vector $y \in H$ such that $f(x) = \langle x, y \rangle$ for all $x \in H$
• Establishes isomorphism between a Hilbert space and its dual space
• Crucial for spectral theory allowing representation of linear functionals as inner products
• Enables the development of spectral theorems for self-adjoint operators

Hilbert space geometry

• Geometric properties of Hilbert spaces extend familiar concepts from Euclidean geometry
• These properties play a crucial role in understanding the behavior of operators in spectral theory
• Geometric intuition from finite-dimensional spaces often carries over to infinite-dimensional Hilbert spaces

Parallelogram law

• States that for any two vectors $x$ and $y$ in a Hilbert space $\|x+y\|^2 + \|x-y\|^2 = 2(\|x\|^2 + \|y\|^2)$
• Characterizes inner product spaces among normed vector spaces
• Implies uniform convexity of Hilbert spaces
• Useful in proving various inequalities and convergence results in spectral theory

Pythagorean theorem

• For orthogonal vectors $x$ and $y$ in a Hilbert space $\|x+y\|^2 = \|x\|^2 + \|y\|^2$
• Generalizes to any finite set of mutually orthogonal vectors
• Fundamental in understanding orthogonal decompositions
• Applies to infinite orthogonal series in the context of Fourier expansions

Cauchy-Schwarz inequality

• States that for any vectors $x$ and $y$ in a Hilbert space $|\langle x, y \rangle| \leq \|x\| \|y\|$
• Equality holds if and only if $x$ and $y$ are linearly dependent
• Crucial for proving continuity of inner product and various operator inequalities
• Generalizes to bounded linear operators via the operator norm

Subspaces in Hilbert spaces

• Subspaces of Hilbert spaces inherit many properties of the parent space
• Understanding subspace structure is crucial for analyzing operators and their spectra
• Decomposition of Hilbert spaces into subspaces plays a key role in spectral theory

Closed subspaces

• Subspaces that contain all their limit points
• Every closed subspace of a Hilbert space is itself a Hilbert space
• Characterized by the property that the orthogonal complement of the orthogonal complement is the subspace itself
• Crucial for defining spectral projections and analyzing operator ranges

Orthogonal complements

• For a subspace $M$ of a Hilbert space $H$, the orthogonal complement $M^\perp$ consists of all vectors orthogonal to every vector in $M$
• Always closed even if $M$ is not closed
• Satisfy $(M^\perp)^\perp = \overline{M}$ (closure of $M$)
• Fundamental in spectral decompositions of self-adjoint operators

Direct sum decomposition

• Any Hilbert space $H$ can be decomposed as $H = M \oplus M^\perp$ for any closed subspace $M$
• Every vector $x \in H$ uniquely expressed as $x = y + z$ with $y \in M$ and $z \in M^\perp$
• Generalizes to infinite direct sums in the context of spectral decompositions
• Crucial for understanding the structure of operator ranges and kernels

Operators on Hilbert spaces

• Linear operators on Hilbert spaces form the core subject of spectral theory
• Properties of these operators often reflect and exploit the underlying Hilbert space structure
• Understanding operator behavior is essential for analyzing spectra and developing spectral decompositions

Bounded linear operators

• Linear transformations $T: H_1 \to H_2$ between Hilbert spaces satisfying $\|Tx\| \leq C\|x\|$ for some constant $C$ and all $x \in H_1$
• Operator norm defined as $\|T\| = \sup_{\|x\|=1} \|Tx\|$
• Continuous linear operators are exactly the bounded linear operators
• Form a Banach algebra with composition as multiplication crucial for spectral theory

Adjoint operators

• For a bounded linear operator $T: H_1 \to H_2$, the adjoint $T^*: H_2 \to H_1$ satisfies $\langle Tx, y \rangle_{H_2} = \langle x, T^*y \rangle_{H_1}$ for all $x \in H_1$, $y \in H_2$
• Existence guaranteed by the Riesz representation theorem
• Properties include $(S+T)^* = S^* + T^*$, $(ST)^* = T^*S^*$, and $(T^*)^* = T$
• Fundamental for defining self-adjoint and normal operators

Self-adjoint operators

• Bounded linear operators $T$ satisfying $T = T^*$
• Spectral theorem for self-adjoint operators forms a cornerstone of spectral theory
• Have real spectra and admit spectral decompositions
• Model many physical observables in quantum mechanics

Hilbert bases

• Hilbert bases provide a way to represent elements of a Hilbert space as infinite series
• These bases generalize the concept of orthonormal bases in finite-dimensional vector spaces
• Understanding Hilbert bases is crucial for developing Fourier analysis in abstract Hilbert spaces

Orthonormal bases

• Countable orthonormal sets $\{e_n\}$ such that their linear span is dense in the Hilbert space
• Every separable Hilbert space possesses an orthonormal basis
• Any vector $x$ can be uniquely expressed as $x = \sum_{n=1}^{\infty} \langle x, e_n \rangle e_n$
• Existence of orthonormal bases characterizes separable Hilbert spaces

Fourier series representation

• Expresses any vector $x$ in terms of an orthonormal basis $\{e_n\}$ as $x = \sum_{n=1}^{\infty} c_n e_n$ where $c_n = \langle x, e_n \rangle$
• Coefficients $c_n$ called Fourier coefficients generalize classical Fourier series
• Convergence of the series holds in the norm topology of the Hilbert space
• Fundamental tool in harmonic analysis and partial differential equations

Parseval's identity

• States that for any vector $x$ and orthonormal basis $\{e_n\}$, $\|x\|^2 = \sum_{n=1}^{\infty} |\langle x, e_n \rangle|^2$
• Generalizes Pythagorean theorem to infinite dimensions
• Implies that the map $x \mapsto (\langle x, e_n \rangle)_{n=1}^{\infty}$ is an isometry onto $\ell^2$
• Crucial for proving completeness of orthonormal systems and analyzing Fourier series

Topological aspects

• Topological properties of Hilbert spaces interplay with their algebraic structure
• Understanding these aspects is crucial for analyzing convergence and compactness in spectral theory
• Many results in spectral theory rely on the interplay between different topologies on Hilbert spaces

Weak vs strong convergence

• Strong convergence $x_n \to x$ if $\|x_n - x\| \to 0$ as $n \to \infty$
• Weak convergence $x_n \rightharpoonup x$ if $\langle x_n, y \rangle \to \langle x, y \rangle$ for all $y$ in the Hilbert space
• Strong convergence implies weak convergence but not vice versa
• Weak convergence often sufficient for many results in spectral theory allowing for more general statements

Compactness criteria

• Bounded sets in infinite-dimensional Hilbert spaces are not compact in the norm topology
• Bounded sets are precompact in the weak topology (Banach-Alaoglu theorem)
• Relatively compact sets characterized by total boundedness (finite $\varepsilon$-nets)
• Compact operators defined as those mapping bounded sets to relatively compact sets crucial in spectral theory

Reflexivity of Hilbert spaces

• Every Hilbert space is reflexive meaning $(H^*)^* \cong H$ canonically
• Implies that closed bounded convex sets are weakly compact
• Ensures existence of minimizers for certain functionals important in variational problems
• Simplifies many arguments in spectral theory and functional analysis

Applications of Hilbert spaces

• Hilbert space theory finds extensive applications across mathematics and physics
• The abstract framework of Hilbert spaces unifies diverse areas of study
• Understanding these applications motivates many developments in spectral theory

Quantum mechanics

• State space of quantum systems modeled as a complex Hilbert space
• Observables represented by self-adjoint operators
• Spectral theorem provides physical interpretation of measurements
• Uncertainty principles derived from Hilbert space inequalities (Cauchy-Schwarz)

Signal processing

• $L^2$ spaces of square-integrable functions serve as Hilbert spaces for signals
• Fourier transform interpreted as a unitary operator on $L^2$
• Sampling theorem and bandwidth limitations analyzed using Hilbert space techniques
• Wavelet analysis utilizes orthonormal bases in $L^2$ spaces

Functional analysis connections

• Hilbert spaces provide concrete examples of many abstract functional analysis concepts
• Spectral theory in Hilbert spaces generalizes to Banach algebras and C*-algebras
• Weak topologies and duality in Hilbert spaces motivate more general functional analytic tools
• Many results first proved for Hilbert spaces later extended to more general settings