Difference between revisions of "Real analysis"
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| − | + | Broadly speaking, '''real analysis''' is the study of the real numbers and its topological properties, sequences and series of real numbers, and properties of real-valued functions. Some properties that are studied in the real numbers are the construction of the real numbers, convergence of sequences, subsets of the plane as metric spaces, limits, notions of [[continuity]], [[Derivative|differentiation]], and [[integration]]. | |
| − | + | A common description of real analysis courses is that real analysis is the formal rigorous study of single-variable [[calculus]] with proofs. This view does have merit to it because most (if not all) of the theorems typically presented to students in courses in single-variable calculus are proven rigorously; however, one should note that courses in real analysis also spend considerable amount of time on pathological examples with little concern for applications, and one also aims to generalize and prove results rather than apply results to calculate numerical answers to exercises as one typically does in calculus. | |
| − | + | == Topics == | |
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| − | == | + | === Construction of the real numbers === |
| − | + | The entirety of real analysis is built upon the real numbers, particularly with the notion of completeness in mind. Intuitively, this is described as the fact that the real numbers <math>\mathbb R</math> lack the existence of any "holes" unlike the rational numbers <math>\mathbb Q</math> (for instance, the set <math>\{x \in \mathbb{Q} \mid x^2 < 2\}</math> has no smallest element in the rational numbers). This property of the real numbers is known as the ''least upper bound property''. | |
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| + | Two particularly known constructions of the real numbers are via [[Cauchy sequences|Cauchy sequence]] and [[Dedekind cuts|Dedekind cut]], both of which take <math>\mathbb{Q}</math> and construct <math>\mathbb{R}</math> as a completion of <math>\mathbb{Q}</math>. | ||
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| + | === Sequences of real numbers === | ||
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| + | A [[sequence]] is a function <math>f:\mathbb{N}\to\mathbb{R}</math>. Conventionally, sequences are typically denoted by the notation <math>(s_n)_{n = k}^{\infty} = (s_k,s_{k + 1},\ldots)</math> where <math>f(n)</math> is denoted by <math>s_n</math>. | ||
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| + | In real analysis, particular attention is paid attention to the convergence and divergence of sequences. Intuitively, the idea of convergence is captured by the notion that the sequence "approaches" some value as <math>n</math> becomes arbitrarily large. Also important is the notion of Cauchy sequences which intuitively describe sequences whose terms become arbitrarily close to each other as <math>n</math> becomes arbitrarily large. | ||
== See also == | == See also == | ||
Revision as of 22:52, 4 November 2022
Broadly speaking, real analysis is the study of the real numbers and its topological properties, sequences and series of real numbers, and properties of real-valued functions. Some properties that are studied in the real numbers are the construction of the real numbers, convergence of sequences, subsets of the plane as metric spaces, limits, notions of continuity, differentiation, and integration.
A common description of real analysis courses is that real analysis is the formal rigorous study of single-variable calculus with proofs. This view does have merit to it because most (if not all) of the theorems typically presented to students in courses in single-variable calculus are proven rigorously; however, one should note that courses in real analysis also spend considerable amount of time on pathological examples with little concern for applications, and one also aims to generalize and prove results rather than apply results to calculate numerical answers to exercises as one typically does in calculus.
Topics
Construction of the real numbers
The entirety of real analysis is built upon the real numbers, particularly with the notion of completeness in mind. Intuitively, this is described as the fact that the real numbers 
lack the existence of any "holes" unlike the rational numbers 
(for instance, the set 
has no smallest element in the rational numbers). This property of the real numbers is known as the least upper bound property.
Two particularly known constructions of the real numbers are via Cauchy sequence and Dedekind cut, both of which take 
and construct 
as a completion of .
Sequences of real numbers
A sequence is a function 
. Conventionally, sequences are typically denoted by the notation 
where 
is denoted by 
.
In real analysis, particular attention is paid attention to the convergence and divergence of sequences. Intuitively, the idea of convergence is captured by the notion that the sequence "approaches" some value as 
becomes arbitrarily large. Also important is the notion of Cauchy sequences which intuitively describe sequences whose terms become arbitrarily close to each other as becomes arbitrarily large.
See also
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