Recent changes

Real Analysis · 23 results added · with a worked companion

Complex Analysis · 41 results added · with a worked companion

Abstract Algebra · 37 results added · with a worked companion

Abstract Algebra · 26 results added · with a worked companion

Linear Algebra · 45 results added · with a worked companion

Linear Algebra · 30 results added · with a worked companion

Combinatorics · 29 results added · with a worked companion

Combinatorics · 34 results added · with a worked companion

Combinatorics · 48 results added · with a worked companion

Combinatorics · 31 results added · with a worked companion

Abstract Algebra · 39 results added · with a worked companion

Category Theory · 63 results added · with a worked companion

Real Analysis · 40 results added · with a worked companion

Real Analysis · 21 results added · with a worked companion

Abstract Algebra · 28 results added · with a worked companion

Abstract Algebra · 26 results added · with a worked companion

Linear Algebra · 28 results added · with a worked companion

Abstract Algebra · 30 results added · with a worked companion

Real Analysis · 16 results added · with a worked companion

Real Analysis · 24 results added · with a worked companion

Category Theory · 35 results added · with a worked companion

Real Analysis · 5 results added

• Definition The five completeness properties of an ordered field: least upper bound, monotone convergence, nested intervals, Bolzano-Weierstrass, and Cauchy completeness
• Theorem For an ordered field the five completeness properties are equivalent, provided the Archimedean property is assumed alongside nested intervals and Cauchy completeness
• False statement FALSE: the nested interval property alone implies the least-upper-bound property
• Example The rational function field ℝ(t) ordered by the eventual sign is an ordered field, worked out
• Example ℝ((t⁻¹)), the formal Laurent series field, is Cauchy complete, non-Archimedean, and lacks the least-upper-bound property

Examples & counterexamples →

Real Analysis · 1 result added

• Counterexample Not every ordered field is Archimedean

Real Analysis · 45 results added · with a worked companion

Category Theory · 85 results added · with a worked companion

Linear Algebra · 31 results added · with a worked companion

Abstract Algebra · 35 results added · with a worked companion

Real Analysis · 38 results added · with a worked companion

Linear Algebra · 38 results added · with a worked companion

Real Analysis · 37 results added · with a worked companion

Linear Algebra · 39 results added · with a worked companion

Combinatorics · 48 results added · with a worked companion

Abstract Algebra · 58 results added · with a worked companion

Combinatorics · 29 results added · with a worked companion

Abstract Algebra · 21 results added · with a worked companion

Abstract Algebra · 35 results added · with a worked companion

Abstract Algebra · 37 results added

• Proposition Free equivalence is an equivalence relation and concatenation respects it
• Definition The word-quotient model F_word(X):=W(X)/∼ with multiplication induced by concatenation
• Theorem F_word(X) is a group under [w][v]=[wv]
• Lemma Formal letters act by mutually inverse permutations on the set of reduced words
• Theorem Every class in W(X)/∼ contains exactly one reduced word
• Theorem The word-quotient group W(X)/∼ satisfies the universal property of the free group on X
• Corollary The generator map X→ W(X)/∼ is injective
• Corollary The word-quotient and reduced-word models are uniquely isomorphic compatibly with X

and 29 others on the page.

Examples & counterexamples →

Abstract Algebra · 47 results added

• Definition Equivariant maps and isomorphisms of group actions
• Definition A free group action has no nonidentity element fixing a point
• Definition The fixed-point sets Xᵍ and X^G of a group action
• Theorem Orbit-stabiliser: G/Gₓ→ G· x, gGₓ↦ g· x, is a well-defined bijection
• Corollary Orbit-stabiliser cardinality: |G· x|=[G:Gₓ] whenever either side is finite, and |G|=|Gₓ| |G· x| for finite G
• Lemma If y=g· x, then G_y=gGₓg⁻¹
• Definition The core Core_G(H)=⋂_g∈ GgHg⁻¹ of a subgroup
• Lemma Core_G(H) is the largest normal subgroup of G contained in H

and 39 others on the page.

Examples & counterexamples →

Real Analysis · 3 results added

• Theorem Peano's form: the normalized Taylor remainder tends to zero
• Corollary A uniform derivative bound gives a uniform Taylor remainder bound
• Remark Scope, endpoint, factorial, and deferred-remainder conventions

Real Analysis · 1 result added

• Example The geometric series represents 1/(1-x) for |x|<1 and re-expands explicitly about every c with |c|<1

Real Analysis · 12 results added

• Theorem Integrable functions on [a,b] form a set closed under sums and scalar multiples, and ∫ₐᵇ(λ f+μ g) = λ∫ₐᵇ f + μ∫ₐᵇ g
• Theorem If f ≤ g on [a,b] and both are integrable then ∫ₐᵇ f ≤ ∫ₐᵇ g; and m(b-a) ≤ ∫ₐᵇ f ≤ M(b-a)
• Theorem For a<c<b: f is integrable on [a,b] if and only if it is integrable on [a,c] and on [c,b], and then ∫ₐᵇ f = ∫ₐᶜ f + ∫_cᵇ f; with the oriented form for arbitrary a,b,c
• Theorem If f is continuous on [a,b] and g is integrable with g ≥ 0, there is ξ ∈ [a,b] with ∫ₐᵇ fg = f(ξ)∫ₐᵇ g
• Theorem The second fundamental theorem: if G is differentiable on [a,b] with G' = f and f is integrable, then ∫ₐᵇ f = G(b)-G(a)
• Theorem If u,v are differentiable on [a,b] with u',v' integrable, then ∫ₐᵇ u v' = u(b)v(b)-u(a)v(a) - ∫ₐᵇ u'v
• Remark Conventions of this page, and which sharpenings of the integral are taken up later in the reading order
• Example The integral test applied to ∑ 1/ι(k+1)ᵖ for rational p>0, cross-checked against the published p-series theorem

and 4 others on the page.

Examples & counterexamples →

Real Analysis · 22 results added

• Definition A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
• Theorem Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
• Lemma Each ‖·‖ₚ is a norm on ℝⁿ, and the induced metrics are exactly d₁, d₂ and d_∞ of the published metric-spaces page
• Lemma The finite and reverse triangle inequalities for a norm; and for n ≥ 1 every norm N on ℝⁿ satisfies N(x) ≤ C‖ x‖₁ and is Lipschitz, hence continuous, for d₂
• Theorem For n ≥ 1 all norms on ℝⁿ are equivalent
• Corollary For n ≥ 1 every bounded sequence in ℝⁿ has a convergent subsequence
• Definition Vector-valued functions f : A → ℝᵐ, their limits and continuity, with the dictionary to the metric notions
• Theorem A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions

and 14 others on the page.

Examples & counterexamples →

Real Analysis · 21 results added

• Definition Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions
• Lemma Uniform convergence of real-valued functions implies pointwise convergence
• Theorem A sequence of real-valued functions converges uniformly if and only if it is uniformly Cauchy
• Definition A series of real-valued functions and its pointwise and uniform convergence through its partial sums
• Corollary A series of real-valued functions converges uniformly if and only if its tails are uniformly small
• Lemma Uniform limits respect sums and scalar multiples
• Lemma Products converge uniformly when both factors converge uniformly and one limiting factor and one approximating family are uniformly bounded
• Theorem The uniform limit of continuous real-valued functions on a metric space is continuous

and 13 others on the page.

Examples & counterexamples →

Real Analysis · 10 results added

• Lemma The exponential series converges absolutely for every real argument
• Corollary The exponential is positive and satisfies exp(-x)=1/exp(x)
• Theorem The exponential function is strictly increasing
• Theorem The exponential tends to +∞ at +∞ and to 0 at -∞
• Corollary The exponential is a continuous bijection from ℝ onto (0,∞)
• Theorem Regular normalized multiplicative Cauchy equations characterize the exponential
• Theorem For every real x, (1+x/n)ⁿ→exp x
• Theorem The number e is irrational

and 2 others on the page.

Examples & counterexamples →

Real Analysis · 2 results added

• Theorem Lebesgue's criterion in ℝᵐ: a bounded function on a closed nondegenerate rectangle is Riemann integrable iff its discontinuity set is null
• Theorem A bounded set in ℝᵐ is Jordan measurable iff its boundary is null, equivalently of content zero

Real Analysis · 4 results added

• Theorem Mertens' theorem: if ∑ aₖ converges absolutely to A and ∑ bₖ converges to B, their Cauchy product converges to AB
• Theorem For pₖ ≥ 0 the product ∏ (1 + pₖ) converges iff ∑ pₖ converges, with 1 + ∑_k<n pₖ ≤ ∏_k<n(1+pₖ) ≤ 1/(1 - ∑_k<n pₖ) when ∑_k<n pₖ < 1; for 0 ≤ pₖ < 1 the product ∏ (1 - pₖ) converges iff ∑ pₖ converges and its partial products tend to 0 otherwise; and ∑ |pₖ| convergent implies ∏ (1+pₖ) convergent
• Remark Selected sums and products on this page that are proved to exist without being evaluated, and what their evaluation waits for
• Counterexample The Cauchy product of ∑_k ≥ 0 (-1)ᵏ/√k+1 with itself has |cₙ| ≥ 1 for every n, so it diverges

Examples & counterexamples →

Real Analysis · 5 results added

• Theorem f is continuous at c ∈ A if and only if f(xₖ) → f(c) for every sequence in A converging to c, the converse direction costing countable choice
• Theorem A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs
• Theorem The image of a compact subset of ℝ under a continuous real function is compact
• False statement FALSE: a continuous real function on a bounded domain attains a greatest value
• Counterexample The identity is uniformly continuous on ℝ and its square is not, so uniform continuity is not preserved by products

Examples & counterexamples →

Topology · 2 results added

• Lemma Finite pointwise minima of continuous maps to [0,1] are continuous
• Corollary Assuming countable choice, perfect normality, and hence T₆, is hereditary

Real Analysis · 6 results added

• Theorem Heine criterion: lim_x → c f(x) = L iff f(xₖ) → L for every sequence in A ∖ {c} converging to c
• Lemma If f has a finite limit at c then f is bounded on some punctured neighbourhood of c
• Theorem Composition of limits holds under either hypothesis: f is defined at L with value M, or g avoids L on a punctured neighbourhood of c
• Remark The classical form of the oscillator above is sin(1/x), which this library can only construct much later
• Counterexample With g ≡ 0 and f equal to 0 off the origin and 1 at it, lim g = 0 and lim_y → 0 f = 0 while f ∘ g ≡ 1
• Counterexample On the domain {0} ∪ [1,2] every real is vacuously a limit at 0

Examples & counterexamples →

Real Analysis · 5 results added

• Definition The oscillation ω_f(S) = sup{ |f(x) - f(y)| : x, y ∈ S } of f on a set and the oscillation ω_f(c) = inf_δ > 0 ω_f(A ∩ N_δ(c)) at a point, both taken in the extended reals
• Theorem Every G_δ subset of ℝ is the set of continuity points of some f : ℝ → ℝ, so the G_δ sets are exactly the continuity sets
• Theorem Semicontinuous extreme value theorem: an upper semicontinuous function on a nonempty compact K ⊆ ℝ is bounded above and attains a maximum, and a lower semicontinuous one is bounded below and attains a minimum
• Theorem Baire's theorem: a Baire class one function on a closed bounded interval [a,b] is continuous at the points of a dense subset of [a,b] that is the trace of a G_δ set, so its set of discontinuities is meager
• False statement FALSE: every additive f : ℝ → ℝ is of the form x ↦ cx for a single real c

Topology · 2 results added

• Lemma Under choice, every open cover of a paracompact Hausdorff space has locally finite open refinements {Vₛ} and {Wₛ} with overlineVₛ⊆ Wₛ⊆overlineWₛ⊆ Uₛ
• Counterexample Assuming choice, two paracompact lower-limit lines can have a nonparacompact product

Examples & counterexamples →

Real Analysis · 7 results added

• Theorem For aₖ, bₖ > 0 with aₖ/bₖ → L: if L ∈ (0,∞) the two series share their behaviour, while L = 0 and L = ∞ give one implication each
• Theorem Root test: limsup |aₖ|^1/k < 1 gives absolute convergence and hence convergence, > 1 gives divergence, and = 1 decides nothing
• Theorem Ratio test: limsup |aₖ₊₁/aₖ| < 1 gives absolute convergence and hence convergence, and liminf |aₖ₊₁/aₖ| > 1 gives divergence
• Remark How the nonnegative tests are ordered by strength, and which of them this page cannot state without the logarithm
• False statement FALSE: if aₖ → 0 then ∑ aₖ converges
• Example The harmonic series ∑ 1/k diverges, by condensation and by Oresme block grouping
• Example Abel-Dini applied to ∑ 1/k: ∑ 1/(k sₖ) still diverges while ∑ 1/(k sₖ²) converges

Examples & counterexamples →

Real Analysis · 6 results added

• Corollary ℚ is F_σ, meager and not G_δ, while the irrationals are G_δ, residual and not F_σ
• Lemma A sequence of intervals covering [a,b] has total length at least b - a, so no interval of positive length has measure zero
• Theorem For a compact subset of ℝ, measure zero and content zero coincide
• Example Which points of [0,1] lie in the Cantor set, read off their ternary expansions, with 1/4 worked out
• Counterexample ℚ is dense in ℝ and has measure zero
• Counterexample The irrationals form a residual G_δ set that is not F_σ

Examples & counterexamples →

Real Analysis · 2 results added

• Theorem On an interval I, for f continuous on I and differentiable at every interior point: f' ≥ 0 throughout gives f nondecreasing, f' > 0 gives f increasing, f' ≤ 0 and f' < 0 give the two decreasing forms; conversely a nondecreasing f has f' ≥ 0 and a nonincreasing f has f' ≤ 0 wherever it is differentiable, and no strict converse is claimed
• Counterexample x ↦ √x on (0,1] is differentiable with unbounded derivative and is not Lipschitz there, so the boundedness hypothesis in the Lipschitz corollary cannot be dropped

Examples & counterexamples →

Real Analysis · 2 results added

• Lemma If m ≤ f ≤ M on [a,b] then m(b-a) ≤ L(f,P) ≤ underline∫ₐᵇ f ≤ overline∫ₐᵇ f ≤ U(f,P) ≤ M(b-a) for every partition P; in particular every constant function is integrable, with ∫ₐᵇ c = c(b-a)
• Example ∫₀³ lfloor x rfloor = 3: the floor function is nondecreasing, hence integrable, and the integral is computed from the uniform partitions

Examples & counterexamples →

Topology · 22 results added

• Lemma Every uniformity has a base of symmetric entourages
• Lemma A metric on a nonempty set generates an entourage uniformity whose induced topology and uniformly continuous maps are the usual metric notions, and this uniformity is separated
• Definition Uniform space in the uniform-cover formulation
• Lemma On a nonempty set, entourage uniformities and uniform-cover structures determine one another
• Definition A gauge of pseudometrics and, on a nonempty set, the uniformity it generates
• Theorem On a nonempty set, entourages and uniform covers give equivalent definitions of a uniform space in ZF, and under dependent choice they are also equivalent to gauges of pseudometrics
• Lemma Every convergent filter on a uniform space is Cauchy
• Lemma A Cauchy filter with a cluster point converges to that point

and 14 others on the page.

Examples & counterexamples →

Topology · 6 results added

• Theorem Urysohn's lemma, under the axiom of dependent choice: in a normal space two disjoint closed sets are separated by a continuous function into [0,1], and conversely such a space is normal
• Theorem Tietze's extension theorem, under dependent choice: a continuous map from a closed subspace of a normal space into [a,b] extends continuously to the whole space, and this property characterises normality
• Remark Which results on this page spend dependent choice, which spend countable choice, and which are theorems of ZF
• Example A Urysohn function for (-∞, 0] and [1, ∞) in ℝ, written down and checked against the definition
• Example In a metric space the function d(x,A)/(d(x,A) + d(x,B)) separates two disjoint closed sets outright, so the metric case spends no choice principle
• Example Sierpinski space is normal and not completely regular, so the T₁ hypothesis in the Urysohn corollary is not decoration

Examples & counterexamples →

Topology · 2 results added

• Theorem A space is Hausdorff if and only if its diagonal is closed in the square carrying the product topology
• Example The graph of a continuous f : ℝ → ℝ is closed in ℝ²

Examples & counterexamples →

Topology · 6 results revised

• Lemma For continuous maps into a convex subset of ℝⁿ, the straight-line formula defines a continuous homotopy · revised
• Theorem Any two continuous maps into a nonempty convex subset of ℝⁿ are homotopic by straight lines · revised
• Corollary Every nonempty convex subset of ℝⁿ is contractible · revised
• Example The formula H(x,t)=(1-t)f(x)+tg(x) gives an explicit homotopy between maps into ℝⁿ · revised
• Example Every nonempty interval and every ℝⁿ with n≥1 contracts to any chosen point · revised
• Example Two paths with the same endpoints in a convex subset of ℝⁿ are path homotopic relative to their endpoints · revised

Examples & counterexamples →

Real Analysis · 2 results added

• Counterexample A sequence with limsup = +∞: the greatest subsequential limit exists only in overlineℝ
• Counterexample Null times divergent has no rule: xₖ = 1/k with yₖ = ck gives product limit c, and with yₖ = k² gives divergence

Real Analysis · 3 results added

• Theorem ℝ((t⁻¹)) is an ordered field, ordered by the sign of the leading coefficient
• Theorem Every Cauchy sequence in ℝ((t⁻¹)) converges: K is sequentially Cauchy complete
• Corollary ℝ((t⁻¹)) has the nested interval property for lengths tending to 0