Principles of Mathematical Analysis¶
Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
Cited by¶
21 citations across 21 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Boundedness
- Boundedness is the structural property that the values, magnitudes, or resource-uses of a set, sequence, function, or process do not exceed some fixed finite threshold — formally, a subset \(S\) of a metric space \((X, d)\) is bounded if there exist a centre \(x_0 \in X\) and a finite radius \(M < \infty\) such that \(d(x, x_0) \leq M\) for every \(x \in S\), with the equivalent formulation that \(S\) has finite diameter \(\operatorname{diam}(S) := \sup\{d(x, y) : x, y \in S\} < \infty\), as Rudin (1976) develops in his standard treatment of metric spaces.
This sourcestandard treatment of metric spaces in which a subset is bounded iff it is contained in a ball of finite radius / has finite diameter.
- Boundedness is the structural property that the values, magnitudes, or resource-uses of a set, sequence, function, or process do not exceed some fixed finite threshold — formally, a subset \(S\) of a metric space \((X, d)\) is bounded if there exist a centre \(x_0 \in X\) and a finite radius \(M < \infty\) such that \(d(x, x_0) \leq M\) for every \(x \in S\), with the equivalent formulation that \(S\) has finite diameter \(\operatorname{diam}(S) := \sup\{d(x, y) : x, y \in S\} < \infty\), as Rudin (1976) develops in his standard treatment of metric spaces.
- Continuity
- 4. Continuity property and its scope: the variant of continuity is named — pointwise continuity at
x₀(the epsilon-delta condition holds at one point), continuity onX(it holds at every point), uniform continuity onX— Rudin (1976)This sourceCanonical undergraduate real-analysis textbook; develops uniform continuity, the classification of discontinuities, the Weierstrass extreme-value theorem, and the Bolzano-Weierstrass theorem within the standard epsilon-delta framework.
- 4. Continuity property and its scope: the variant of continuity is named — pointwise continuity at
- Dense Set
- A is dense in B (with respect to a notion of closeness) exactly when the closure of A equals B
This sourceStandard reference defining density (closure equals the host), the density and countability of the rationals in the uncountable reals, and the ε-neighbourhood characterization.
- A is dense in B (with respect to a notion of closeness) exactly when the closure of A equals B
- Enthymeme
- Consider a proof that writes "since \(f\) is continuous on the compact set \(K\), it attains its maximum" and stops.
This sourceStandard analysis text establishing the extreme value theorem (a continuous function on a compact set attains its maximum), the suppressed premise in the worked proof example.
- Consider a proof that writes "since \(f\) is continuous on the compact set \(K\), it attains its maximum" and stops.
- Higher Order Function
- The composition algebra over rules is fully present and load-bearing: \(D\) composes with itself (\(D^2\) is the second derivative), has an approximate inverse (integration, via the fundamental theorem of calculus), and obeys distributive-style laws (linearity: \(D(af + bg) = aD(f) + bD(g)\); the product and chain rules govern composition).
This sourceEstablishes linearity of the derivative, the product and chain rules, and the fundamental theorem of calculus relating differentiation and integration — the composition algebra over the derivative operator.
- The composition algebra over rules is fully present and load-bearing: \(D\) composes with itself (\(D^2\) is the second derivative), has an approximate inverse (integration, via the fundamental theorem of calculus), and obeys distributive-style laws (linearity: \(D(af + bg) = aD(f) + bD(g)\); the product and chain rules govern composition).
- Quantifier
- To say \(\lim_{x \to a} f(x) = L\) is to assert: for every \(\varepsilon > 0\), there exists a \(\delta > 0\), such that for all \(x\) with \(0 < |x-a| < \delta\), we have \(|f(x)-L| < \varepsilon\).
This sourceStandard source for the ε–δ definition of a limit and for the pointwise-versus-uniform continuity distinction, which turns entirely on quantifier order.
- To say \(\lim_{x \to a} f(x) = L\) is to assert: for every \(\varepsilon > 0\), there exists a \(\delta > 0\), such that for all \(x\) with \(0 < |x-a| < \delta\), we have \(|f(x)-L| < \varepsilon\).
Domain-specific¶
- Absolute convergence
- Closed Set
- Closedness interacts with boundedness to produce the Heine–Borel theorem: in ℝⁿ a set is compact — the condition that guarantees continuous functions attain their extreme values, that sequences have convergent subsequences, and that the extreme-value theorem holds — if and only if it is both closed and bounded
This sourceStates the Heine-Borel equivalence of closed-and-bounded with compactness in R*, and the attainment of a maximum by a continuous function on a compact set (Theorem 4.16). Rudin's Heine-Borel and Weierstrass theorems, the pair the sentence's Bolzano-Weierstrass reference points to. Rudin's maximum-attainment theorem for continuous functions on compact sets, the result the Weierstrass label attaches to.
Supported in partVerified against the work's full text
“[Heine]-Borel theorem. 2.41. Theorem”
- Closedness interacts with boundedness to produce the Heine–Borel theorem: in ℝⁿ a set is compact — the condition that guarantees continuous functions attain their extreme values, that sequences have convergent subsequences, and that the extreme-value theorem holds — if and only if it is both closed and bounded
- Compactness
- Equivalently, in metric spaces, a space is compact if and only if every sequence has a convergent subsequence (sequential compactness); in ℝⁿ, Heine–Borel gives a more concrete characterisation — a subset is compact if and only if it is both closed and bounded
This sourceRudin's theorem that a continuous mapping of a compact metric space into a metric space is uniformly continuous. Rudin (1976) treats equicontinuous families of functions and their uniform continuity, the immediate content on which the Arzela-Ascoli characterisation rests.
Supported in partVerified against a saved copy of the source
“Every bounded sequence in R* contains a convergent subsequence”
- Equivalently, in metric spaces, a space is compact if and only if every sequence has a convergent subsequence (sequential compactness); in ℝⁿ, Heine–Borel gives a more concrete characterisation — a subset is compact if and only if it is both closed and bounded
- Complete field
- Conditional convergence
- Decimal representation
- Derivative
- Its algebraic structure is governed by the sum rule, product rule, quotient rule, and chain rule — the last being the statement that (f∘g)′(x) = f′(g(x))·g′(x)
This sourceThe text checked of Rudin supports the chain rule and its formula; it does not show the sum, product and quotient rules for derivatives.
Supported in partVerified against the work's full text
“the difference/—^, the product/^, and the [quotient]//^ of the two functions, with the understanding”
- Its algebraic structure is governed by the sum rule, product rule, quotient rule, and chain rule — the last being the statement that (f∘g)′(x) = f′(g(x))·g′(x)
- Directional Derivative
- Function series
- Local boundedness
- Mean of a function
- Ratio Test
- Real-valued function
- Root Test
- Uniform norm
Verification¶
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