PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Analysis ↓


Real Analysis


Real analysis is a branch of mathematics that deals with the set of real numbers and the functions defined on them. This field is fundamental to understanding the behavior of spaces and functions and provides the basis for more advanced topics in mathematics. Real analysis is essential in broad areas such as calculus, sequences, series, and continuity and integration of functions. Here, we will explore the basic ideas and elements of real analysis, and try to make them understandable for anyone starting their journey into this deep field of study.

Real number

Real numbers are the basis of real analysis. They include all rational numbers, such as integers and fractions, and irrational numbers, such as the square root of 2 or the number π (pi). The set of real numbers is represented by ℝ.

An important property of real numbers is that they are complete. This means that the real number line has no "gaps," which is necessary for defining concepts like limits and continuity.

0 - + 0.5 1

Scenes

A sequence is an ordered list of numbers that is usually defined by some rule. Mathematicians often work with sequences to understand the properties of relationships between different numbers. For example, the sequence of natural numbers is

1, 2, 3, 4, 5, 6, ...

A sequence may converge to a specific number, known as its limit. If a sequence converges, it is the value that the terms of the sequence approach as the index of the sequence goes to infinity. For example, for the sequence

a_n = 1/n

As n approaches infinity, the sequence will converge to 0.

n=1 n=2 n=3 n=4 n=5 , Limit = 0

Series

Simply put, a series is a sum of sequences. If you have a sequence, you can get a series by adding each term of the sequence together:

a_1 + a_2 + a_3 + ... = ∑ a_n

You can think of it as infinite summation - the process of adding an endless sequence of numbers. A well-known series is the geometric series, which is defined as:

∑ ar^n

where a is the first term, and r is the common ratio. An example might be when a = 1 and r = 1/2:

1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2

This series converges to 2, which means that the sum of this infinite series approaches 2 the more terms you add.

Functions and continuity

Functions are essential objects in real analysis. They map elements from one set to another and often take real numbers as input and provide real numbers as output—an important relationship that helps explain the behavior of mathematical models.

A function f(x) is continuous at a point a if the limit of the function as x approaches a is equal to the function value f(a):

lim (x → a) f(x) = f(a)
f(a) A

In simple terms, a function is continuous where you can draw it without lifting your pencil from the paper. Continuity is important because many functions useful in applications are continuous, and this property ensures that small changes in the input lead to only small changes in the output.

Differentiation and integration

Differentiation is an important concept in calculus, and thus in real analysis, which is concerned with understanding how functions change. If you have a function f(x), its derivative is often denoted as f'(x) and tells you how f(x) changes as x changes.

A simple visual example involves using derivatives to find the slope of a curve at a point. For a simple function like f(x) = x^2, the derivative is

f'(x) = 2x

Differentiation helps in identifying the critical points of a function where it can have maximum or minimum values.

Meanwhile, integration is the process of finding the integral of a function, which can be understood as the "area under the curve" of that function. If you have a function f(x), its integral is usually represented as:

∫ f(x) dx

In the case of f(x) = x^2, the integral is

∫ x^2 dx = (1/3)x^3 + C

where C is the constant of integration. Integration is fundamental to many applications in physics, engineering, and beyond, including calculating area, volume, and other quantities.

Conclusion

Real analysis is a deep and essential part of mathematics that deals with real numbers and functions. It helps to understand and work with many concepts fundamental to calculus and other advanced mathematical topics. By studying real analysis, you build an essential foundation for exploring complex mathematical theories and solving complex problems.

The concepts of real numbers, sequences, series, continuity, differentiation, and integration are only the surface of this amazing subject. Going deeper into real analysis will reveal the depth and complexity of mathematics, which provides a very structured way of expressing and handling real-world problems in a logical framework.


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Real Analysis

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