Real Analysis

Royden, H. L., & Fitzpatrick, P. M. (2010). Real Analysis. Pearson.

Type

Book

Intellectual base

Textbook

Year

2010

Link

https://openlibrary.org/isbn/9780131437470

Cited from

mathematics

Cited by

4 citations across 4 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Boundedness
  • Boundedness is not the same as completeness, a structural distinction Royden and Fitzpatrick (2010) develop carefully in their treatment of metric and Banach spaces.

    This sourcedevelops completeness (Cauchy-sequence convergence, e.g., of Lp spaces) and boundedness as independent properties of metric and Banach spaces.

• Measure
  • In mathematics and analysis the Lebesgue measure gives rigorous meaning to the size of arbitrary subsets of the line, enabling integration far beyond what Riemann sums allow.

    This sourceConstructs Lebesgue measure on the real line from length on intervals via Carathéodory's extension theorem and develops the Lebesgue integral, including the measure-zero rationals.

Domain-specific

• Function series
  • … transfers from A power series sums functions a_n(x-x0)^n and converges to a function inside its radius, often uniformly on compact subintervals. to Analysis never substitutes pointwise for uniform convergence when interchanging limits, integrals or derivatives and states exceptional sets for almost-everywhere claims..
• Helly's selection theorem
  • … identity and a convenient observable transfers from Uniformly bounded distribution functions admit a pointwise convergent subsequence at continuity points of a limiting distribution function. to Probability theory extracts a weakly convergent subsequence of measures through their cumulative distribution functions..

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