Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Functional Analysis ↓


Hilbert Spaces


Hilbert space is a beautiful topic in functional analysis that lies at the intersection of algebra, geometry, and calculus. It provides a framework for discussing infinite-dimensional vector spaces endowed with inner products. This topic forms the backbone of many classical and modern analysis theories.

Introduction to vector spaces

Before we dive into Hilbert spaces, it is important to understand what a vector space is. A vector space over a field F (such as the real numbers ℝ) is a set V equipped with two operations: vector addition and scalar multiplication. The operations must satisfy certain axioms such as associativity, commutativity, identity elements, and inverses.

Inner product space

An inner product space is a vector space with an additional structure called the inner product. This inner product allows you to define angles and lengths in a vector space, similar to the dot product in Euclidean space.

The inner product of two vectors u and v in a vector space V is usually denoted by ⟨u, v⟩ and satisfies the following properties:

1. Conjugate symmetry: ⟨u, v⟩ = ⟨v, u⟩
2. Linearity in the first argument: ⟨au + bw, v⟩ = a⟨u, v⟩ + b⟨w, v⟩
3. Positive definiteness: ⟨v, v⟩ ≥ 0, and ⟨v, v⟩ = 0 if and only if v = 0
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Defining a Hilbert space

A Hilbert space is an inner product space that is complete, meaning that all Cauchy sequences in the space converge to a limit within the space. In other words, every sequence of vectors that "looks" like it should converge actually converges to a point in the space.

Completeness is important because it allows the application of many analytical techniques. When a space is incomplete, certain natural limits and sums within that space may not exist.

Examples of Hilbert spaces

Real coordinate space ℝ n

The simplest example of a Hilbert space is the real n-dimensional space ℝ n with the standard dot product:

⟨u, v⟩ = u 1 v 1 + u 2 v 2 + ... + u n v n

Here, the space ℝ n is complete and every vector can be expressed in terms of n coordinates. This is the most familiar and elementary example of a Hilbert space.

Point 1Point 2Hey

The space of square integrable functions L 2 (a, b)

Another important example is the space of square-integrable functions on the interval [a, b]. This is often the space of functions f(x) such that:

∫ a b |f(x)| 2 dx < ∞

In this space the inner product is defined as:

⟨f, g⟩ = ∫ a b f(x)g(x) dx

This space is complete and is a Hilbert space, which plays an essential role in quantum mechanics and other areas of physics and engineering.

To see this, imagine functions defined on graphs within the interval [a, b], which are similar to waves or curves. The space considers these functions to be finite through their square integrals.

Properties of Hilbert space

Hilbert spaces have several important properties that make them essential tools in analysis and related fields. Here are some key properties:

Orthogonality

Two vectors u and v in a Hilbert space are orthogonal if their inner product is zero:

⟨u,v⟩ = 0

Orthogonal vectors lie at right angles to each other in space. This concept helps simplify many problems, mainly because non-zero orthogonal vectors are linearly independent.

Orthonormal basis

The orthonormal basis of a Hilbert space is a set of vectors that are mutually orthogonal, each of unit length, and span the space.

If {e 1 , e 2 , ..., e n } is an orthonormal basis for a subspace of a Hilbert space, then any vector v in the space can be uniquely written as:

V = a 1 e 1 + a 2 e 2 + ... + a n e n

where the coefficients are given by the inner product:

a i = ⟨v, ei ⟩

Projection theorem

An important result in the theory of Hilbert spaces is the projection theorem. It states that for any vector v and a closed subspace S, there exists a unique vector u in S such that:

v = u + w

where w is perpendicular to S In other words, every vector in the space can be decomposed into the sum of a vector in the subspace S and a vector perpendicular to S

Applications of Hilbert space

Hilbert spaces are important in a variety of fields due to their versatility and completeness. Some application areas include:

Quantum mechanics

In quantum mechanics, the state of a quantum system is described by vectors in a Hilbert space. Physical observables are represented by operators on these spaces, and the concepts of superposition and probability amplitudes are naturally expressed within this framework.

Signal processing

Signal processing and Fourier analysis rely heavily on Hilbert spaces. They provide a way to break down signals into sinusoidal components, analyze frequencies, and perform efficient data conversion.

Machine learning

In machine learning, kernel methods leverage inner product spaces to interpolate data into high-dimensional spaces, facilitating complex classification and regression that are otherwise challenging in standard Euclidean spaces.

Conclusion

The study of Hilbert spaces in functional analysis lays the groundwork for further explorations in mathematics and physics. By providing a complete and robust structure for infinite-dimensional spaces, they allow rigorous treatment of many complex phenomena.

Understanding Hilbert spaces opens the door to many advanced topics, making them an invaluable part of the mathematical toolbox.


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