Key Concepts
For more details on this topic, see Construction of the real numbers.The foundation of real analysis is the construction of the real numbers from the rational numbers. This is usually carried out by Dedekind–MacNeille completion, Dedekind cuts, or by completion of Cauchy sequences. Key concepts in real analysis are filters, nets, real sequences and their limits, convergence, continuity, differentiation, and integration. Real analysis is also used as a starting point for other areas of analysis, such as complex analysis, functional analysis, and harmonic analysis, as well as for motivating the development of topology, and as a tool in other areas, such as applied mathematics.
Important results include the Bolzano–Weierstrass and Heine–Borel theorems, the intermediate value theorem and mean value theorem, the fundamental theorem of calculus, and the monotone convergence theorem.
Various ideas from real analysis can be generalized from real space to general metric spaces, as well as to measure spaces, Banach spaces, and Hilbert spaces.
Read more about this topic: Real Analysis
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—Johann Wolfgang Von Goethe (1749–1832)
“It is impossible to dissociate language from science or science from language, because every natural science always involves three things: the sequence of phenomena on which the science is based; the abstract concepts which call these phenomena to mind; and the words in which the concepts are expressed. To call forth a concept, a word is needed; to portray a phenomenon, a concept is needed. All three mirror one and the same reality.”
—Antoine Lavoisier (1743–1794)