Alphabeta MathArticle items · 01 — Cauchy-Schwarz inequality for $L^2$
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Holder's inequality for integrals, including the endpoint cases
- Equality in Holder's inequality for 1 < p < ∞
- Euclidean spheres and closed balls as subspaces of ℝⁿ
- For a nonempty subset of ℝⁿ with n≥1, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent
- 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
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
- For n≥2, the punctured space ℝⁿ∖{0} is polygonally connected
- Radial normalisation x↦ x/‖ x‖₂ is continuous on ℝⁿ∖{0}
- A finite concatenation of straight segments in ℝⁿ is a continuous path
- Every path-connected space is connected, and every path component lies inside a component
- Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure
- Every smooth vector bundle admits a smooth bundle metric
- For a metric space with its metric topology, compactness in the topological sense is compactness in the metric sense, and the two notions of compact subset coincide
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- For n ≥ 1 the product topology on n copies of the usual topology of ℝ is the metric topology of d_∞ on ℝⁿ, and hence also of d₁ and d₂, so ℝⁿ as a product and ℝⁿ as a metric space are one space
- ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it
- The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- Simplicial and singular homology agree
- A simplex has zero reduced simplicial homology
- Augmentation at 0-simplices and reduced singular homology
- The singular chain complex and singular homology
- The singular chain homotopy formula
- A chain homotopy
- Chain-homotopic maps induce the same map on homology
- For every positive-sized square matrix over a commutative ring, Aadj(A)=adj(A)A=det(A)I
- A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit
- In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided
- The exterior derivative squares to zero
- Cartan's magic formula
- Cauchy's mean value theorem: for f, g continuous on [a,b] with a<b and differentiable on (a,b) there is c ∈ (a,b) with (f(b)-f(a))g'(c) = (g(b)-g(a))f'(c); no hypothesis on g' is needed in this product form
- The derivative f'(c) = lim_x → c f(x) - f(c)/x - c of f : A → ℝ at a point c ∈ A that is a limit point of A, and differentiability on a set
- Continuity of f : A → ℝ at a point of A and on A: the ε-δ condition, its agreement with lim_x → c f(x) = f(c) at a limit point, and continuity at an isolated point
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- The ε-δ limit lim_x → c f(x) = L of f : A → ℝ at a limit point c of A
- The double has a well-defined smooth structure
- A smooth tensor field
- A smooth differential k-form
- Smooth functions on relatively open half-space sets
- Transverse embedded submanifolds
- The transverse preimage theorem
- Codimension and hypersurfaces
- The disc formula for the volume of a solid of revolution
- Jordan inner and outer content and Jordan measurable bounded sets in ℝᵐ
- For a natural n ≥ 1 the function x ↦ xⁿ is differentiable everywhere with derivative ι(n) x^ n-1; for n = 0 it is the constant 1, with derivative 0; for a natural n ≥ 1 the function x ↦ x⁻ⁿ is differentiable at every x ≠ 0 with derivative -ι(n) x⁻ⁿ⁻¹; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
- 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)
- Choice function
- De rham cochain complex
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- Finite, countably infinite, countable, uncountable
- Open cover, subcover, compact metric space, and compact subset of a metric space
- ℂ=ℝ[x]/(x²+1) is a field, every element is uniquely a+bi, and every nonzero element has inverse (a-bi)/(a²+b²)
- Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0}
- The Cauchy-sequence reals have the least-upper-bound property
- Continuity of a map of topological spaces at a point and globally
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- A point lies in the closure of A iff every basic neighbourhood of it meets A; the closure is the smallest closed superset and equals A together with its derived set
- Separated sets, disconnection, and connected subset of ℝ
- Cohomology object of a cochain complex
- Cʳ and smooth maps between smooth manifolds
- Smooth maps are continuous
- An open subset of a smooth manifold has a canonical restricted smooth structure
- Existence uniqueness and smooth dependence of geodesics
- The Axiom of Countable Choice (AC_ω)
- Smooth manifolds and their smooth charts
- The Lie bracket of smooth vector fields
- Coordinate formula for the Lie bracket
- Compatible orientation classes over compact subsets
- Excision for singular homology
- Local Hölder and scaled C-two-alpha norms on balls
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Bounded Cᵏ domains and boundary charts
- Bounded C1 domains and their outward normals
- The tangent bundle as a disjoint union
- Smooth sections, local sections, and support
- Interior product on alternating covectors
- Local coordinate expression for a differential form
- Gradient hessian and divergence connection formulas
- Riemannian divergence
- The Lie derivative of a tensor field
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
CorollaryLibrary prerequisite
Cauchy-Schwarz inequality for L²
CorollaryLibrary prerequisite
For n≥1, every Euclidean closed ball and every Euclidean sphere of positive radius is compact
CorollaryLibrary prerequisite
For n≥2, the sphere Sⁿ⁻¹ is path-connected and connected
CorollaryLibrary prerequisite
Assuming countable choice, every smooth manifold admits a Riemannian metric
CorollaryLibrary prerequisite
A subset of ℝⁿ with the product topology is compact exactly when it is closed and bounded, the product topology being the Euclidean metric topology
CorollaryLibrary prerequisite
Homology of spheres
CorollaryLibrary prerequisite
Homotopic maps induce the same map on singular homology
CorollaryLibrary prerequisite
If det(A) is a unit, then A⁻¹=det(A)⁻¹adj(A)
CorollaryLibrary prerequisite
Lie derivative commutes with the exterior derivative
CorollaryLibrary prerequisite
The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
CorollaryLibrary prerequisite
Smooth functions and tensor fields extend locally across the boundary
CorollaryLibrary prerequisite
Transverse embedded submanifolds intersect in the expected codimension
CorollaryLibrary prerequisite
A closed three-dimensional ball of radius r≥0 has volume 4π r³/3
DefinitionLibrary prerequisite
The Axiom of Choice
DefinitionLibrary prerequisite
Closed and exact differential forms
DefinitionLibrary prerequisite
Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
DefinitionLibrary prerequisite
Real and imaginary parts, complex conjugation, and modulus
DefinitionLibrary prerequisite
Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
DefinitionLibrary prerequisite
De rham cohomology
DefinitionLibrary prerequisite
Diffeomorphisms and local diffeomorphisms of manifolds
DefinitionLibrary prerequisite
Domain and exponential map of a connection
DefinitionLibrary prerequisite
Embedded submanifolds and slice charts
DefinitionLibrary prerequisite
The exterior derivative by the invariant vector-field formula
DefinitionLibrary prerequisite
Fundamental class of a compact oriented manifold
DefinitionLibrary prerequisite
Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
DefinitionLibrary prerequisite
Interior product of a form by a vector field
DefinitionLibrary prerequisite
Laplace–Beltrami operator as the trace of the Hessian
DefinitionLibrary prerequisite