Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Real Analysis ↓


Functional Analysis


Functional analysis is a branch of mathematical analysis that deals with the study of vector spaces and operators acting on them. It is characterized by the use of function spaces and is closely connected with the study of infinite-dimensional spaces. The subject has deep applications in many branches of mathematics as well as physics, economics, and engineering.

Basic concepts

Let's start with some basic concepts that are essential to understanding functional analysis.

1. Vector space

A vector space over a field ( F ) (usually the real numbers ( mathbb{R} ) or the complex numbers ( mathbb{C} )) is a collection of objects called vectors that can be added together and multiplied by scalars from ( F ), subject to certain axioms.

Example:
Consider the set of all real sequences: 
( ell^{infty} = { x = (x_1, x_2, ldots) | x_i in mathbb{R} } )
This set forms a vector space where addition and scalar multiplication are defined componentwise.

2. Standard location

A normed space is a vector space ( V ) over a field ( F ) with a norm function ( |cdot| : V to mathbb{R} ) that satisfies:

  • ( |x| geq 0 ) and ( |x| = 0 iff x = 0 ) (Positivity)
  • ( |alpha x| = |alpha||x| ) for all ( alpha in F ) and ( x in V ) (Isomorphism)
  • ( |x + y| leq |x| + |y| ) (Triangle Inequality)
Example:
The space with Euclidean norm ( mathbb{R}^n )
( |x| = sqrt{x_1^2 + x_2^2 + cdots + x_n^2} )
There is a standard location.

3. Banach space

A Banach space is a complete normed space; that is, a normed space (V) in which every Cauchy sequence converges to an element in (V). These spaces are important in functional analysis because of their analytic properties.

B Inside a ball in Banach space

In the above illustration, the idea of completeness is represented by the assumption that all sequences inside the ball must meet at a point inside it.

4. Hilbert spaces

A Hilbert space is a special type of Banach space where the norm is obtained from an inner product. The inner product ( langle cdot , cdot rangle : V times V to mathbb{R} ) (or ( mathbb{C} )) satisfies:

  • ( langle x, x rangle geq 0 ), and ( langle x, x rangle = 0 iff x = 0 )
  • ( langle x, y rangle = overline{langle y, x rangle} ) (conjugate symmetry)
  • ( langle x + y, z rangle = langle x, z rangle + langle y, z rangle ) (linearity in the first argument)
Example:
The space ( L^2(mathbb{R}) ), the set of all square-integrable functions:
The function ( f: mathbb{R} to mathbb{C} ) where
( int_{-infty}^{infty} |f(x)|^2 , dx < infty )
is a Hilbert space with inner product 
( langle f, g rangle = int_{-infty}^{infty} f(x)overline{g(x)} , dx ).

Operators on spaces

Operators play a central role in functional analysis. These are mappings between function spaces that preserve the vector space structure.

1. Linear operators

A linear operator between vector spaces ( T: V to W ) is a function that satisfies:

  • ( T(x + y) = T(x) + T(y) ) for all ( x, y in V )
  • ( T(alpha x) = alpha T(x) ) for all ( alpha in F ) and ( x in V )
Example:
Consider ( T: mathbb{R}^2 to mathbb{R}^2 ) given by
( T(x, y) = (2x, 3y) ).
It is a linear operator because
( T((x_1, y_1) + (x_2, y_2)) = T(x_1 + x_2, y_1 + y_2) = (2(x_1 + x_2), 3(y_1 + y_2)) )
And
( T(alpha(x, y)) = T(alpha x, alpha y) = (2alpha x, 3alpha y) = alpha T(x, y) ).

2. Bounded and unbounded operators

A linear operator ( T: V to W ) between normed spaces is bounded if there exists a constant ( C geq 0 ) such that ( |T(x)|_W leq C |x|_V ) for all ( x in V ). If no such ( C ) exists, the operator is said to be unbounded.

Example:
The differentiation operator is given by ( D: C^infty(mathbb{R}) to C^infty(mathbb{R}) )
( d(f) = f' )
is unbounded because for any continuous function ( f ), 
Increasing ( |f|_{L^infty} ) does not limit the size of ( f' ).

Operator's spectrum

The spectrum of a bounded linear operator (T) on a Banach space is a set of scalars that generalizes the concept of eigenvalues for matrices. It can be classified into three types: point, continuous, and residual spectrum.

1. Eigenvalue

The point spectrum consists of all ( lambda in F ) such that ( T - lambda I ) is not injective. These are the eigenvalues of ( T ).

Example:
Consider the operator ( T: mathbb{R}^2 to mathbb{R}^2 ) given by the matrix
( begin{pmatrix} 3 & 0  0 & 2 end{pmatrix} ). 
The eigen values are the solutions of the following \( det(begin{pmatrix} 3 & 0 \ 0 & 2 end{pmatrix} - lambda I) = 0 ),
The yield is ( lambda = 3, 2 ).
3 2

The above simple visualization illustrates the idea of finding eigenvalues geometrically by interpreting it as a rescaling of dimensions.

Duality in functional analysis

Duality is an essential concept that refers to the relationship between a vector space and its dual space.

1. Dual sights

The dual space ( V^* ) of a vector space ( V ) is the set of all linear functionals on ( V ). A linear functional ( phi: V to F ) maps every vector to a scalar in the field ( F ), satisfying the linearity conditions.

Example:
For ( V = mathbb{R}^n ), every functional can be written as 
( phi(x) = a_1x_1 + a_2x_2 + cdots + a_nx_n ),
where ( a_i ) are constant real numbers.

2. Rees representation theorem

This theorem states that in a Hilbert space (H), every bounded linear functional (f) can be represented as an inner product with a fixed element in (H):

Suppose ( f in H^* ). Then there exists a unique ( y in H ) such that 
( f(x) = langle x, y rangle ) for all ( x in H ).
H H*

The visualization shows how elements of a Hilbert space ( H ) are projected into its dual space ( H^* ) via linear functions.

Applications and further studies

Functional analysis is indispensable in quantum mechanics, statistics, biological models, and economic theories. It provides tools for understanding limits, approximations, and other phenomena in infinite-dimensional settings.

  • In quantum mechanics, the states of a system are described by vectors in a Hilbert space.
  • In optimization, understanding duality problems can lead to more efficient algorithms.
  • In signal processing, the Fourier transform is a linear operation in function space.

Conclusion

Functional analysis is a rich and deep field, full of abstract but applicable insights. It combines algebra, geometry, calculus, and logic to explore the spaces and operators that underlie the functioning of many complex systems. Whether studying spaces directly or understanding their dualities and operators, this corner of mathematics remains an active and vibrant area of research.


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