Absolute value in an ordered field

Depends on

• Ordered field

Used by

• If f,g are integrable on [a,b] then so are | f|, f², fg, max(f,g) and min(f,g), and |∫ₐᵇ f| ≤ ∫ₐᵇ| f| Corollary
• ℝ((t⁻¹)) has the nested interval property for lengths tending to 0 Corollary
• {q ∈ ℚ : q ≥ 0, q² < 2} is closed and bounded in ℚ and is not compact Counterexample
• ‖·‖₁ on ℝ² violates the parallelogram law, so no symmetric bilinear form induces it Counterexample
• A function that is not Riemann integrable although | f| is Counterexample
• A sequence with limsup = +∞: the greatest subsequential limit exists only in ℝ̄ Counterexample
• In {0} ∪ [1,2] with the metric of ℝ, the closure of B(0,1) = {0} is {0} while the closed ball is {0,1} Counterexample
• On (0,∞) the metrics |x-y| and |1/x - 1/y| have the same topology and are not uniformly equivalent Counterexample
• On ℝ the metrics |x-y| and min(|x-y|,1) are uniformly but not Lipschitz equivalent Counterexample
• Refuted: a pointwise bounded family of continuous functions is equicontinuous. The spikes are bounded by 1 everywhere and are not equicontinuous at 0 Counterexample
• Refuted: C(X,Y) is closed in the topology of pointwise convergence. The ramps on [0,1] converge pointwise to a discontinuous limit Counterexample
• Refuted: convergence uniformly on every compact subset of ℝ implies uniform convergence. The maps x ↦ x/(n+1) separate the two Counterexample
• The sign function is Riemann integrable on [-1,1] and has no primitive there Counterexample
• xₖ = 1 + (-1)ᵏ, yₖ = 1 + (-1)ᵏ⁺¹ give limsup(xₖ yₖ) = 0 < 4 Counterexample
• ℤ and {n + 1/n : n ≥ 2} are disjoint closed subsets of ℝ at distance 0, so the set-to-set distance is not a metric Counterexample
• A norm on a real vector space, the induced metric, and the dictionary with the metric axioms Definition
• Absolute and conditional convergence of improper integrals Definition
• Absolute continuity on a compact interval Definition
• Absolutely convergent and conditionally convergent series, and the general starting index Definition
• Bounded variation and total variation on an interval Definition
• Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ Definition
• For n≥1, the determinant over a commutative ring by the Leibniz formula, and |det A| for a real matrix Definition
• Normwise and componentwise backward error for an approximate linear-system solution Definition
• Permutation matrices, partial pivoting, and the pivot-growth factor Definition
• Sequences, convergence, Cauchyness, monotonicity, boundedness and closed intervals in an arbitrary ordered field Definition
• The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices Definition
• The p-norms ‖ x‖ₚ for rational p ≥ 1, and ‖ x‖_∞ Definition
• The standard relative floating-point model and unit roundoff Definition
• The topology of compact convergence on C(X,Y) for metric X and Y: uniform convergence on each compact subset of X Definition
• The ε-neighbourhood and the punctured ε-neighbourhood of a point of ℝ Definition
• (-1)ᵏ has liminf = -1 and limsup = 1, so it does not converge Example
• |x| is Lipschitz and absolutely continuous but not C¹ on [-1,1] Example
• A positive sequence making all three inequalities of the ratio-to-root chain strict Example
• aₖ = 2^-k + (-1)ᵏ has liminf aₖ₊₁/aₖ = 1/8, limsup aₖ₊₁/aₖ = 2 and lim aₖ^1/k = 1/2 Example
• Dini's theorem applied to a nondecreasing sequence of piecewise linear approximations on [0,1], and what fails when the limit is not continuous Example
• For every k≥0, xᵏ|x| is Cᵏ but not Cᵏ⁺¹ Example
• min(|x-y|, 1) on ℝ has the usual topology and diameter at most 1 Example
• The 1-Lipschitz maps of a metric space into ℝ form a uniformly equicontinuous family, and the distance functions x ↦ d(x,A) all belong to it Example
• The block sequence 1/1; 1/2, 2/2; 1/3, 2/3, 3/3; … has subsequential limit set exactly [0,1] Example
• The comparison constants between ‖·‖₁, ‖·‖₂ and ‖·‖_∞ on ℝ², and vectors attaining each Example

…and 51 more results.

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Sources

• W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
• T. Tao, Analysis I, 3rd ed. (standard reference, not scraped)
• Dartmouth College analysis notes: Absolute value (standard reference, not scraped)