1.4 Normed spaces

Normed spaces are fundamental to spectral theory, providing a framework for measuring distances and defining convergence in vector spaces. They extend vector spaces by introducing a norm function, enabling the analysis of linear operators and their properties.

Understanding normed spaces is crucial for studying operator behavior in spectral theory. Key concepts include p-norms, supremum norms, and the topology induced by norms. These ideas form the foundation for exploring linear operators, dual spaces, and applications in quantum mechanics and functional analysis.

Definition of normed spaces

• Normed spaces form a crucial foundation for spectral theory by providing a mathematical framework to study linear operators and their properties
• These spaces extend the concept of vector spaces by introducing a norm, which allows for measuring distances and defining convergence
• Understanding normed spaces is essential for analyzing the behavior of operators in spectral theory and their applications in quantum mechanics and functional analysis

Vector spaces vs normed spaces

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• Vector spaces consist of a set of vectors and scalar multiplication operations
• Normed spaces add a norm function that assigns a non-negative real number to each vector
• The norm function $\|x\|$ satisfies three key properties:
  • Non-negativity: $\|x\| \geq 0$ for all vectors x, with $\|x\| = 0$ if and only if $x = 0$
  • Homogeneity: $\|\alpha x\| = |\alpha| \|x\|$ for all scalars α and vectors x
  • Triangle inequality: $\|x + y\| \leq \|x\| + \|y\|$ for all vectors x and y
• Normed spaces enable the definition of distance between vectors as $d(x,y) = \|x - y\|$

Properties of norms

• Norms induce a metric topology on the vector space
• Continuity of vector addition and scalar multiplication follows from the properties of norms
• The reverse triangle inequality holds: $|\|x\| - \|y\|| \leq \|x - y\|$
• Norms satisfy the parallelogram identity in inner product spaces
• Different norms can be equivalent, leading to the same topology on the vector space

Types of norms

p-norms

• p-norms generalize the concept of length to higher dimensions
• For a vector $x = (x_1, \ldots, x_n)$, the p-norm is defined as $\|x\|_p = (\sum_{i=1}^n |x_i|^p)^{1/p}$ for $1 \leq p < \infty$
• Common p-norms include:
  • 1-norm (Manhattan norm): $\|x\|_1 = \sum_{i=1}^n |x_i|$
  • 2-norm (Euclidean norm): $\|x\|_2 = \sqrt{\sum_{i=1}^n |x_i|^2}$
• p-norms satisfy Hölder's inequality: $|\langle x, y \rangle| \leq \|x\|_p \|y\|_q$ where $\frac{1}{p} + \frac{1}{q} = 1$

Supremum norm

• Also known as the maximum norm or infinity norm
• Defined as $\|x\|_\infty = \max_{1 \leq i \leq n} |x_i|$ for finite-dimensional spaces
• For function spaces, it becomes $\|f\|_\infty = \sup_{x \in X} |f(x)|$
• Useful in analyzing uniform convergence and boundedness of functions
• Satisfies the property $\|x\|_\infty \leq \|x\|_p$ for all $p \geq 1$

Euclidean norm

• Most commonly used norm in physics and engineering applications
• Defined as $\|x\|_2 = \sqrt{\sum_{i=1}^n |x_i|^2}$ for finite-dimensional spaces
• In inner product spaces, it can be expressed as $\|x\| = \sqrt{\langle x, x \rangle}$
• Satisfies the Cauchy-Schwarz inequality: $|\langle x, y \rangle| \leq \|x\| \|y\|$
• Invariant under orthogonal transformations, making it useful in rotational symmetries

Topology in normed spaces

Open and closed sets

• Open sets in normed spaces contain an open ball around each point
• Closed sets contain all their limit points
• The complement of an open set is closed, and vice versa
• Boundary points belong to the closure of both a set and its complement
• Important for defining continuous functions and studying convergence properties

Convergence in normed spaces

• A sequence $(x_n)$ converges to x if $\lim_{n \to \infty} \|x_n - x\| = 0$
• Cauchy sequences satisfy $\lim_{m,n \to \infty} \|x_m - x_n\| = 0$
• Convergence in norm implies pointwise convergence but not vice versa
• Different norms may lead to different notions of convergence
• Weak convergence defined using linear functionals: $x_n \rightharpoonup x$ if $f(x_n) \to f(x)$ for all continuous linear functionals f

Completeness and Banach spaces

• A normed space is complete if every Cauchy sequence converges
• Complete normed spaces are called Banach spaces
• Completeness ensures the existence of limits for certain sequences and series
• Examples of Banach spaces include:
  • $\ell^p$ spaces of p-summable sequences
  • $L^p$ spaces of p-integrable functions
• Completeness plays a crucial role in fixed point theorems and operator theory

Linear operators on normed spaces

Bounded linear operators

• Linear operators T satisfy $T(\alpha x + \beta y) = \alpha T(x) + \beta T(y)$ for all scalars α, β and vectors x, y
• Bounded operators satisfy $\|T(x)\| \leq M\|x\|$ for some constant M and all vectors x
• Continuity of linear operators equivalent to boundedness
• Bounded linear operators form a vector space
• The set of bounded linear operators between two normed spaces is itself a normed space

Operator norm

• Defined as $\|T\| = \sup_{\|x\| \leq 1} \|T(x)\|$ for a bounded linear operator T
• Satisfies properties similar to vector norms:
  • Non-negativity: $\|T\| \geq 0$, with $\|T\| = 0$ if and only if T is the zero operator
  • Homogeneity: $\|\alpha T\| = |\alpha| \|T\|$ for all scalars α
  • Triangle inequality: $\|S + T\| \leq \|S\| + \|T\|$ for operators S and T
• Submultiplicative property: $\|ST\| \leq \|S\| \|T\|$ for composable operators S and T
• Used to study convergence of operator sequences and series

Equivalence of norms

Equivalent norms

• Two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a vector space V are equivalent if there exist positive constants c and C such that $c\|x\|_1 \leq \|x\|_2 \leq C\|x\|_1$ for all x in V
• Equivalent norms induce the same topology on the vector space
• Convergence in one norm implies convergence in the equivalent norm
• Completeness is preserved under equivalent norms
• Useful for transferring properties between different normed spaces

Finite-dimensional spaces

• All norms on a finite-dimensional vector space are equivalent
• This equivalence stems from the compactness of the unit sphere in finite dimensions
• Allows for the interchangeable use of different norms in finite-dimensional analysis
• Simplifies the study of linear operators on finite-dimensional spaces
• Does not hold for infinite-dimensional spaces, where different norms can lead to distinct topologies

Dual spaces

Continuous linear functionals

• Linear functionals are linear maps from a vector space to its scalar field
• Continuous linear functionals satisfy $|f(x)| \leq M\|x\|$ for some constant M and all vectors x
• The set of all continuous linear functionals on a normed space X forms its dual space X*
• Dual spaces play a crucial role in functional analysis and optimization theory
• Riesz representation theorem relates continuous linear functionals to elements of the space in certain cases (Hilbert spaces)

Dual norm

• Defined on the dual space X* as $\|f\|_{X^*} = \sup_{\|x\| \leq 1} |f(x)|$
• Satisfies properties similar to the original norm
• Allows for the study of weak* topology on the dual space
• Hahn-Banach theorem guarantees the existence of continuous linear functionals with specific properties
• Bidual space (X*)* and its relationship to X important in reflexivity studies

Geometric properties

Convexity in normed spaces

• A set C in a normed space is convex if for any x, y in C and t in [0,1], tx + (1-t)y is also in C
• Convex sets have important properties in optimization and functional analysis
• Normed spaces themselves are convex sets
• Closed balls and open balls in normed spaces are convex
• Intersection of convex sets is convex, allowing for the definition of convex hulls

Parallelogram law

• Holds in inner product spaces: $\|x+y\|^2 + \|x-y\|^2 = 2(\|x\|^2 + \|y\|^2)$ for all vectors x and y
• Characterizes inner product spaces among normed spaces
• Fails to hold in general normed spaces that are not inner product spaces
• Related to the notion of uniform convexity in Banach spaces
• Important in the study of Hilbert spaces and their geometric properties

Normed algebras

Banach algebras

• Algebras equipped with a submultiplicative norm: $\|xy\| \leq \|x\| \|y\|$ for all elements x and y
• Complete with respect to the norm, forming a Banach space
• Examples include:
  • The algebra of bounded linear operators on a Banach space
  • The algebra of continuous functions on a compact space with the supremum norm
• Spectral theory of Banach algebras generalizes many results from matrix theory
• Gelfand theory relates commutative Banach algebras to spaces of continuous functions

C*-algebras

• Banach algebras with an involution operation * satisfying $(x^*)^* = x$ and $\|x^*x\| = \|x\|^2$
• Generalize the algebra of bounded linear operators on a Hilbert space
• Examples include:
  • Matrix algebras with the operator norm and conjugate transpose as involution
  • The algebra of bounded operators on a Hilbert space
• Important in quantum mechanics and noncommutative geometry
• Gelfand-Naimark theorem represents abstract C*-algebras as concrete operator algebras

Applications in spectral theory

Spectrum in normed algebras

• The spectrum of an element x in a normed algebra A is the set of complex numbers λ such that x - λe is not invertible (e is the unit element)
• Spectral radius defined as $r(x) = \sup\{|\lambda| : \lambda \in \sigma(x)\}$
• Spectral radius formula: $r(x) = \lim_{n \to \infty} \|x^n\|^{1/n}$
• Spectrum is always non-empty for elements of a Banach algebra
• Generalizes the concept of eigenvalues from matrix theory to infinite-dimensional operators

Resolvent set

• The resolvent set of an element x is the complement of its spectrum
• For λ in the resolvent set, the resolvent operator $(x - \lambda e)^{-1}$ exists and is bounded
• Resolvent identity: $R(\lambda) - R(\mu) = (\mu - \lambda)R(\lambda)R(\mu)$ where R(λ) is the resolvent operator
• Analytic properties of the resolvent function crucial in spectral theory
• Used to study the spectral properties of linear operators in functional analysis

Approximation in normed spaces

Best approximation theorem

• States that in a Hilbert space H, for any closed subspace M and any point x in H, there exists a unique point y in M such that $\|x - y\| = \inf_{z \in M} \|x - z\|$
• This unique point y is the orthogonal projection of x onto M
• Generalizes to uniformly convex Banach spaces with some modifications
• Forms the basis for many optimization algorithms and approximation methods
• Applications in signal processing, machine learning, and numerical analysis

Density and separability

• A subset A of a normed space X is dense if its closure is the whole space X
• A normed space is separable if it contains a countable dense subset
• Examples of separable spaces include:
  • $\ell^p$ spaces for $1 \leq p < \infty$
  • $L^p$ spaces on finite measure spaces for $1 \leq p < \infty$
• Separability important in the study of weak topologies and in proving existence theorems
• Dense subspaces allow for approximation of elements in the larger space