1.1 Vector spaces

Vector spaces form the foundation of linear algebra and spectral theory. They provide a framework for studying linear transformations and their properties, crucial for understanding eigenvalues and eigenvectors in analyzing linear operators.

This topic covers the definition and axioms of vector spaces, examples and non-examples, subspaces, bases, dimensions, linear transformations, inner product spaces, norms, completeness, quotient spaces, dual spaces, and direct sums. These concepts are essential for characterizing and manipulating vector spaces in spectral theory.

Definition of vector spaces

• Vector spaces form the foundation of linear algebra and spectral theory, providing a framework for studying linear transformations and their properties
• In spectral theory, vector spaces are crucial for understanding eigenvalues and eigenvectors, which are central to analyzing linear operators

Axioms of vector spaces

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• Closure under addition ensures the sum of any two vectors in the space remains within the space
• Associativity of addition allows for flexible grouping of vector additions
• Commutativity of addition means the order of vector addition does not affect the result
• Existence of zero vector provides a neutral element for vector addition
• Existence of additive inverse enables subtraction and negation of vectors
• Closure under scalar multiplication maintains that scaling a vector keeps it within the space
• Distributivity of scalar multiplication over vector addition allows for factoring out common terms
• Distributivity of scalar addition over vector multiplication enables simplification of complex expressions
• Associativity of scalar multiplication ensures consistent results when scaling vectors multiple times
• Existence of multiplicative identity (1) preserves vectors under scalar multiplication

Examples of vector spaces

• $\mathbb{R}^n$ represents n-dimensional real coordinate space, fundamental in many applications
• Function spaces (continuous functions, polynomials) allow for analysis of infinite-dimensional vector spaces
• Matrix spaces $M_{m,n}(\mathbb{R})$ encompass all m×n matrices with real entries
• Sequence spaces ($\ell^p$ spaces) consist of infinite sequences with specific summability properties
• Polynomial spaces $P_n$ include all polynomials of degree at most n

Non-examples of vector spaces

• The set of positive real numbers fails to be a vector space due to lack of additive inverses
• The set of integers is not closed under scalar multiplication with real numbers
• The unit circle in $\mathbb{R}^2$ lacks closure under addition and scalar multiplication
• The set of 2×2 invertible matrices is not closed under addition

Vector subspaces

• Subspaces are essential in spectral theory for analyzing invariant subspaces of linear operators
• Understanding subspaces helps in decomposing complex vector spaces into simpler, more manageable components

Criteria for subspaces

• Non-emptiness ensures the subspace contains at least the zero vector
• Closure under addition maintains that the sum of any two vectors in the subspace remains in the subspace
• Closure under scalar multiplication guarantees that scaling any vector in the subspace keeps it within the subspace
• These criteria are both necessary and sufficient for a subset to be a subspace

Span of vectors

• The span of a set of vectors consists of all linear combinations of those vectors
• Spans generate subspaces, providing a way to construct subspaces from given vectors
• Minimal spanning sets lead to the concept of basis
• The dimension of a span is at most the number of vectors in the spanning set

Linear independence vs dependence

• Linear independence occurs when no vector in a set can be expressed as a linear combination of the others
• Linear dependence implies at least one vector can be written as a linear combination of the others
• Independence is crucial for determining bases and dimensions of vector spaces
• Dependent sets contain redundant information and can be reduced without losing span

Basis and dimension

• Bases and dimensions are fundamental concepts in spectral theory, essential for characterizing vector spaces
• These concepts allow for finite representations of infinite-dimensional spaces in many cases

Definition of basis

• A basis is a linearly independent set of vectors that spans the entire vector space
• Every vector in the space can be uniquely expressed as a linear combination of basis vectors
• Bases provide a coordinate system for the vector space
• The number of vectors in a basis determines the dimension of the space

Properties of basis

• All bases of a given vector space have the same number of elements
• Any linearly independent set can be extended to a basis
• Any spanning set can be reduced to a basis
• Bases allow for unique representations of vectors in terms of coordinates

Dimension of vector spaces

• The dimension of a vector space is the number of vectors in any basis of the space
• Finite-dimensional spaces have a finite number of basis vectors
• Infinite-dimensional spaces (function spaces) have bases with infinitely many vectors
• The dimension is an invariant property of a vector space, independent of the choice of basis

Coordinate representations

• Coordinates express vectors as linear combinations of basis vectors
• The coordinate vector contains the coefficients of the linear combination
• Coordinate transformations describe changes between different bases
• Coordinates enable computational methods for solving linear algebra problems

Linear transformations

• Linear transformations are central to spectral theory, as they preserve the vector space structure
• These transformations form the basis for studying operators in quantum mechanics and other fields

Definition and properties

• Linear transformations preserve vector addition and scalar multiplication
• They map zero to zero and respect linear combinations of vectors
• Linearity allows for matrix representation in finite-dimensional spaces
• Composition of linear transformations results in another linear transformation

Kernel and image

• The kernel (null space) consists of all vectors mapped to zero by the transformation
• The image (range) includes all vectors that are outputs of the transformation
• The rank-nullity theorem relates the dimensions of kernel, image, and domain
• Injective transformations have a trivial kernel, while surjective ones have a full image

Matrix representation

• Matrices represent linear transformations in finite-dimensional spaces
• The columns of the matrix are the images of the standard basis vectors
• Matrix multiplication corresponds to composition of linear transformations
• Change of basis formulas involve similarity transformations of matrices

Inner product spaces

• Inner product spaces extend vector spaces with a notion of angle and length
• These spaces are crucial in spectral theory for studying self-adjoint operators and their spectra

Definition and properties

• Inner products are symmetric, bilinear (or sesquilinear) forms satisfying positive definiteness
• They induce a norm and metric on the vector space
• The Cauchy-Schwarz inequality bounds the inner product by the product of norms
• Parallelogram law and polarization identity relate inner products to norms

Orthogonality and orthonormality

• Vectors are orthogonal if their inner product is zero
• Orthonormal sets consist of mutually orthogonal unit vectors
• The Gram-Schmidt process constructs orthonormal bases from arbitrary bases
• Orthogonal complements partition spaces into mutually perpendicular subspaces

Gram-Schmidt process

• Converts a linearly independent set of vectors into an orthonormal basis
• Proceeds by successively orthogonalizing each vector against previously orthogonalized ones
• Normalization ensures each resulting vector has unit length
• The process is numerically unstable and often requires reorthogonalization in practice

Normed vector spaces

• Normed spaces generalize the notion of length to abstract vector spaces
• These spaces are fundamental in functional analysis and spectral theory

Definition of norms

• Norms assign non-negative real numbers to vectors, satisfying positive definiteness
• They are absolutely homogeneous, scaling linearly with scalar multiplication
• The triangle inequality ensures the norm of a sum is bounded by the sum of norms
• Norms induce metrics, allowing for concepts of continuity and convergence

Examples of norms

• The p-norms ($\ell^p$ norms) generalize Euclidean distance to higher dimensions
• The supremum norm ($\ell^\infty$ norm) measures the maximum absolute component
• The Euclidean norm ($\ell^2$ norm) corresponds to the usual notion of length
• Matrix norms include the Frobenius norm and operator norms

Equivalence of norms

• Two norms are equivalent if they induce the same topology on the vector space
• In finite-dimensional spaces, all norms are equivalent
• Equivalence implies the existence of positive constants relating the norms
• Norm equivalence preserves completeness and other topological properties

Completeness in vector spaces

• Completeness ensures that Cauchy sequences converge within the space
• This property is crucial for spectral theory, especially in infinite-dimensional spaces

Cauchy sequences

• Cauchy sequences have elements that get arbitrarily close to each other
• In complete spaces, all Cauchy sequences converge to a limit within the space
• Completeness allows for powerful theorems like the Banach Fixed Point Theorem
• Incomplete spaces can be completed by adding limit points of Cauchy sequences

Banach spaces

• Banach spaces are complete normed vector spaces
• They allow for the extension of many finite-dimensional results to infinite dimensions
• The Hahn-Banach theorem guarantees the existence of certain linear functionals
• Banach spaces include $\ell^p$ spaces, $L^p$ spaces, and spaces of continuous functions

Hilbert spaces

• Hilbert spaces are complete inner product spaces
• They combine the algebraic structure of inner product spaces with topological completeness
• The Riesz representation theorem relates linear functionals to inner products
• Separable Hilbert spaces have countable orthonormal bases (Schauder bases)

Quotient spaces

• Quotient spaces allow for the construction of new vector spaces by identifying equivalent elements
• These spaces are important in spectral theory for studying operators modulo compact perturbations

Definition and construction

• Quotient spaces are formed by partitioning a vector space into equivalence classes
• The equivalence relation is defined by a subspace, identifying vectors that differ by elements of the subspace
• Equivalence classes become the elements of the quotient space
• Vector space operations are well-defined on these equivalence classes

Properties of quotient spaces

• The dimension of a quotient space is the difference between the dimensions of the original space and the subspace
• The canonical projection map from the original space to the quotient space is linear and surjective
• The kernel of the projection map is the subspace used to define the equivalence relation
• Quotient spaces inherit many properties from the original space (e.g., completeness)

Isomorphism theorems

• The First Isomorphism Theorem relates quotient spaces to images of linear transformations
• It states that the quotient of a vector space by the kernel of a linear transformation is isomorphic to the image
• The Second Isomorphism Theorem deals with the interaction of subspaces and quotient spaces
• The Third Isomorphism Theorem allows for "factoring" of quotient spaces

Dual spaces

• Dual spaces consist of all linear functionals on a vector space
• They play a crucial role in spectral theory, particularly in the study of adjoint operators

Definition of dual space

• The dual space $V^*$ consists of all linear maps from V to the scalar field
• Elements of the dual space are called linear functionals or covectors
• The dual space is itself a vector space with pointwise operations
• In infinite-dimensional spaces, the dual space can be larger than the original space

Dual basis

• For a given basis of V, the dual basis consists of linear functionals that evaluate to 1 on one basis vector and 0 on others
• The dual basis spans the dual space (for finite-dimensional spaces)
• Coordinates with respect to a basis correspond to values of dual basis functionals
• The dual basis allows for easy computation of linear functionals

Reflexivity of spaces

• A space is reflexive if it is naturally isomorphic to its double dual (V ≅ V**)
• All finite-dimensional spaces are reflexive
• Some infinite-dimensional spaces (e.g., Hilbert spaces) are reflexive
• Reflexivity is important in functional analysis and the study of weak topologies

Direct sum and product spaces

• Direct sums and products allow for the construction of larger vector spaces from smaller ones
• These constructions are fundamental in decomposing spaces and operators in spectral theory

Definition and properties

• Direct sums combine vector spaces by taking the Cartesian product with componentwise operations
• The dimension of a direct sum is the sum of the dimensions of the component spaces
• Direct products generalize direct sums to possibly infinite collections of vector spaces
• Both constructions preserve many properties of the component spaces (e.g., completeness)

External vs internal direct sum

• External direct sum constructs a new space from given vector spaces
• Internal direct sum decomposes a single vector space into the sum of its subspaces
• An internal direct sum requires that the subspaces have trivial pairwise intersections
• Every vector in an internal direct sum has a unique representation as a sum of components from the subspaces

Decomposition of spaces

• Direct sum decompositions allow for the analysis of complex spaces in terms of simpler components
• Spectral decomposition expresses a space as a direct sum of eigenspaces
• The primary decomposition theorem decomposes a space into generalized eigenspaces
• Orthogonal decompositions in inner product spaces split spaces into mutually perpendicular subspaces