Real analysis

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.Every nonempty subset of R {displaystyle mathbb {R} } that has an upper bound has a least upper bound that is also a real number. ( a n ) = ( a n ) n ∈ N = ( a 1 , a 2 , a 3 , ⋯ ) {displaystyle (a_{n})=(a_{n})_{nin mathbb {N} }=(a_{1},a_{2},a_{3},cdots )} . a 1 ≤ a 2 ≤ a 3 ≤ … {displaystyle a_{1}leq a_{2}leq a_{3}leq ldots } or a 1 ≥ a 2 ≥ a 3 ≥ … {displaystyle a_{1}geq a_{2}geq a_{3}geq ldots } f ( x ) → L     as     x → x 0 {displaystyle f(x) o L { ext{as}} x o x_{0}} , or lim x → x 0 f ( x ) = L {displaystyle lim _{x o x_{0}}f(x)=L} . a n → a     as     n → ∞ {displaystyle a_{n} o a { ext{as}} n o infty } , or lim n → ∞ a n = a {displaystyle lim _{n o infty }a_{n}=a} ; In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. The theorems of real analysis rely intimately upon the structure of the real number line. The real number system consists of a set ( R {displaystyle mathbb {R} } ), together with two binary operations denoted + and ⋅, and an order denoted <. The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique complete ordered field, in the sense that any other complete ordered field is isomorphic to it. Intuitively, completeness means that there are no 'gaps' in the real numbers. In particular, this property distinguishes the real numbers from other ordered fields (e.g., the rational numbers Q {displaystyle mathbb {Q} } ) and is critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often conveniently expressed as the least upper bound property (see below). There are several ways of formalizing the definition of the real numbers. Modern approaches consist of providing a list of axioms, and a proof of the existence of a model for them, which has above properties. Moreover, one may show that any two models are isomorphic, which means that all models have exactly the same properties, and that one may forget how the model is constructed for using real numbers. Some of these constructions are described in the main article. The real numbers have several important lattice-theoretic properties that are absent in the complex numbers. Most importantly, the real numbers form an ordered field, in which sums and products of positive numbers are also positive. Moreover, the ordering of the real numbers is total, and the real numbers have the least upper bound property: These order-theoretic properties lead to a number of important results in real analysis, such as the monotone convergence theorem, the intermediate value theorem and the mean value theorem. However, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects. In particular, many ideas in functional analysis and operator theory generalize properties of the real numbers – such generalizations include the theories of Riesz spaces and positive operators. Also, mathematicians consider real and imaginary parts of complex sequences, or by pointwise evaluation of operator sequences. Many of the theorems of real analysis are consequences of the topological properties of the real number line. The order properties of the real numbers described above are closely related to these topological properties. As a topological space, the real numbers has a standard topology, which is the order topology induced by order < {displaystyle <} . Alternatively, by defining the metric or distance function d : R × R → R ≥ 0 {displaystyle d:mathbb {R} imes mathbb {R} o mathbb {R} _{geq 0}} using the absolute value function as d ( x , y ) = | x − y | {displaystyle d(x,y)=|x-y|} , the real numbers become the prototypical example of a metric space. The topology induced by metric d {displaystyle d} turns out to be identical to the standard topology induced by order < {displaystyle <} . Theorems like the intermediate value theorem that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than in R {displaystyle mathbb {R} } only. Often, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods. A sequence is a function whose domain is a countable, totally ordered set. The domain is usually taken to be the natural numbers, although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices.