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Article items · 03 — Strongly continuous one-parameter unitary group
Presents 29 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Hilbert space
- A bounded linear operator between normed spaces
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Orthogonality and the orthogonal complement
- Continuity of a map between metric spaces, at a point and globally, in the ε-δ form
- Banach-valued simple function and integral
- Measure-null sets and almost-everywhere statements relative to a measure
- Distributions form a sheaf
- Cᵏ maps and multi-index derivative notation in Euclidean space
- The standard flat function is smooth and flat at zero
- A linear map L:ℝᵐ→ℝⁿ in Euclidean coordinates
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Vector-valued functions f : A → ℝᵐ, their limits and continuity, with the dictionary to the metric notions
- 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
- Projection valued measure
- Pvm integral is a star homomorphism
- Unbounded linear operators: domain, graph and extension
- The Axiom of Countable Choice (AC_ω)
- Scalar and complex measures from a pvm
- Bounded borel pvm integral
- Dominated convergence
- Integer-order Sobolev spaces and their norms
- The Sobolev norm descends to equivalence classes
- Paths, path-connected spaces and path components
- Every path-connected space is connected, and every path component lies inside a component
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- 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
- 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
- A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice
- ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it
- The p-norms ‖ x‖ₚ for rational p ≥ 1, and ‖ x‖_∞
- Each ‖·‖ₚ is a norm on ℝⁿ, and the induced metrics are exactly d₁, d₂ and d_∞ of the published metric-spaces page
- Vector space over a field
- Continuity of a map of topological spaces at a point and globally
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not
- The absolute value makes ℝ a metric space: d(x,y) = |x-y| is a metric, its open balls are the intervals (x-r, x+r), and it is unbounded
- Open ball, closed ball and sphere in a metric space
- 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
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- A compact subset of a metric space is closed and bounded
- |d(x,A) - d(y,A)| ≤ d(x,y), so the distance to a fixed nonempty set is 1-Lipschitz
- Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent
- Open cover, subcover, compact metric space, and compact subset of a metric space
- A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
- The closure of a nonempty A is {x : d(x,A) = 0}, equals A together with its limit points, and is the smallest closed superset
- Interior, closure, boundary, limit point, isolated point and dense subset of a metric space
- Greatest lower bound (infimum)
- Epsilon characterisation of the infimum
- Maximum and minimum of a set
- Isometry, isometric embedding, and the subspace metric on a subset
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- The one-dimensional endpoint estimate on a bounded interval
- One-dimensional W^1,p functions have unique absolutely continuous representatives
- The ACL characterisation of W^1,p
- Weak Leibniz rule with a smooth factor
- Compactly supported Sobolev functions extend by zero in every integer order
- Compactly supported smooth functions are dense in W^k,p(Rⁿ)
- Holder's inequality for integrals, including the endpoint cases
- The Axiom of Choice
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- The recursion theorem
- Bochner-integrable function
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- 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
- The countable-choice principle used in the foliation pair
- A function is a relation f with (a,b) ∈ f and (a,c) ∈ f implying b = c; f : A → B, the value f(a), domain and codomain
- Real-order Hˢ as weighted Fourier distributions
- Japanese-bracket and Laplacian Bessel-potential operators
- Real powers of the Japanese bracket act on Schwartz space
- Distributional derivatives are polynomial Fourier multipliers
- Smooth polynomially bounded multipliers on schwartz space
- Regular distribution from a locally integrable function
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
- A Euclidean bump for a compact set inside an open set
- Test function inclusion in schwartz space is continuous
- Fourier transform is a topological automorphism of Schwartz space
- Real powers for positive bases, with the zero-base positive-exponent convention
- The exponential function is strictly increasing
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- Schwartz derivatives are integrable
- Bochner integrability criterion
- The Banach-valued simple integral is well defined
- Additivity of the nonnegative Lebesgue integral
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Linearity, locality, and commutation of weak derivatives
- The Lᵖ norm descends to the quotient and makes Lᵖ a normed space for 1 ≤ p ≤ ∞
- Complex Holder, Minkowski, and the quotient norm
- Integral over a measurable subset
- The essential supremum of a measurable function with respect to a measure
- Complex Lp classes and Euclidean test-function conventions
- A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage
- Lebesgue outer measure is at most Jordan outer content, and a bounded Jordan measurable set is Lebesgue measurable with Lebesgue measure equal to its Jordan content
- The support of a function on ℝⁿ and its compactly supported Riemann integral
- Weak derivative of a locally integrable function
- Uniqueness of a weak derivative as an almost-everywhere class
- Distributional differentiation is continuous and commutes
- Real and imaginary parts, complex conjugation, and modulus
- ℂ=ℝ[x]/(x²+1) is a field, every element is uniquely a+bi, and every nonzero element has inverse (a-bi)/(a²+b²)
- The Cauchy-sequence reals have the least-upper-bound property
- Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0}
- Squaring is monotone on the nonnegatives
- Squares of nonzero elements are positive
- Complex Lp completeness and almost-everywhere subsequences
- Complex finite-simple and smooth compact-support density for finite p
- The complex L² pairing on equivalence classes
- The complex L² pairing is well-defined and satisfies Cauchy–Schwarz
- Complex norm and complex normed space
- Realification of a complex vector space by restriction of scalars
- Complex modulus restricts to real absolute value
- Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed
- Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- Measures are monotone
- Mean value inequality for a differentiable Banach-valued curve
- Fréchet derivative between Banach spaces
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- The induced length is a norm
- Real and complex inner-product spaces and their induced length
- Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ
- Infinitesimal generator of a unitary group
- Bochner dominated convergence theorem
- Bounded linear maps commute with Bochner integration
- Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative
- A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral
- Monotone convergence for the integral
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- Vector addition and scalar multiplication are continuous in a normed space
- Limits and Cauchy sequences of reals
- The rationals embed densely in the reals
- Nonnegativity of a metric is a consequence of the other axioms, not an axiom
- Absolute value in an ordered field
- Basic properties of the absolute value
- The multiplicative identity is positive
- Order is preserved by adding a constant and by adding inequalities
- Inverses of positives are positive, and reciprocation reverses order
- Sign rules for products and monotonicity of multiplication
- Field
- Ordered field
- Complete ordered field (least-upper-bound property)
- Evolution triples and the parabolic energy space
- Time mollification preserves the evolution-space derivative
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
DefinitionLibrary prerequisite
Strongly continuous one-parameter unitary group
DefinitionLibrary prerequisite
Strongly measurable Banach-valued function
DefinitionLibrary prerequisite
Support of a distribution
DefinitionLibrary prerequisite
Test function space d of an open set
DefinitionLibrary prerequisite
The standard smooth step function
DefinitionLibrary prerequisite
The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
DefinitionLibrary prerequisite
Integral of a measurable function against a projection-valued measure
DefinitionLibrary prerequisite
Zero-boundary Sobolev space as a norm closure
ExampleLibrary prerequisite
Every convex subset of ℝⁿ, in particular every ball and ℝⁿ itself, is path-connected and hence connected
ExampleLibrary prerequisite
The distance from a point to a nonempty compact set is attained at a point of that set, and two disjoint compact sets are at positive distance
ExampleLibrary prerequisite
The trace of a one-dimensional Sobolev function is the pair of endpoint values
LemmaLibrary prerequisite
AC supplies the countable and dependent choices used in Banach integration
LemmaLibrary prerequisite
Average convergence for a continuous Banach-valued function
LemmaLibrary prerequisite
The Axiom of Choice implies countable choice
LemmaLibrary prerequisite
Bessel potentials shift Sobolev order
LemmaLibrary prerequisite
Bochner integral norm inequality
LemmaLibrary prerequisite
Bounded restriction and cutoff localisation in Sobolev spaces
LemmaLibrary prerequisite
The published Riemann change-of-variables theorem already gives the Lebesgue formula for continuous compactly supported integrands
LemmaLibrary prerequisite
Classical derivatives agree with weak derivatives
LemmaLibrary prerequisite
Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
LemmaLibrary prerequisite
Complex completeness, density, and inner product: the consumer interface
LemmaLibrary prerequisite
Realification preserves a complex norm and its metric
LemmaLibrary prerequisite
Euclidean balls have positive finite Lebesgue measure
LemmaLibrary prerequisite
Fundamental theorem of calculus for Banach-valued continuous curves
LemmaLibrary prerequisite
The inner product is jointly continuous
LemmaLibrary prerequisite
Laplace resolvents of a unitary group
LemmaLibrary prerequisite
Linearity of the Bochner integral
LemmaLibrary prerequisite
A sequence in a metric space has at most one limit
LemmaLibrary prerequisite