Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Vector addition and scalar multiplication are continuous in a normed space
Statement
Let be a normed space over or . Then
and
are continuous for the product topology on the domain and the norm topology on the target.
Facts & Assumptions
Given: Scalars and vectors .
The scalar field on this page is either with its usual absolute value or with its modulus, and the same norm estimates are valid in both cases (Real and complex scalar conventions for normed spaces).
The norm is -Lipschitz for its metric, so in particular for all vectors and (The reverse triangle inequality in a normed space).
Proof
For addition, by [L2].
For scalar multiplication, , so by the triangle inequality and absolute homogeneity from [L1] and [L2].
Given , step 1.1 shows that if and , then lies in the -ball around ; this is continuity of addition at .
For scalar multiplication at , first require and then choose so that ; step 1.2 then gives whenever and .
Since the point was arbitrary, step 2.1 proves continuity of addition everywhere, and step 2.2 proves continuity of scalar multiplication everywhere.
Depends on
- Real and complex scalar conventions for normed spaces
- The reverse triangle inequality in a normed space
- Continuity of a map of topological spaces at a point and globally
- The product set $\prod_{i \in I} X_i$ 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
- The absolute value makes $\mathbb{R}$ 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
- $\mathbb C$ is the real coordinate plane, with coordinate arithmetic
Used by
- Compact linear operator Definition
- A circle representation with an averaged orthogonal weight form Example
- A half-circle configuration loop traces an elementary half twist Example
- A pure two-strand full twist as an ordered loop Example
- Cutoff extension across a puncture in complex dimension two Example
- Point pushing one puncture around another Example
- The two-point unordered cover of the plane and the monodromy of a half turn Example
- The claim that the ball and the polydisc are biholomorphic False statement
- A bounded-below operator has closed range Lemma
- Bounded finite rank operators are compact Lemma
- Compositions with a compact operator are compact Lemma
- Forgetting the last n points is locally trivial with fibre Fₙ of the punctured manifold Lemma
- Linear combinations of compact operators are compact Lemma
- Linearity of the Bochner integral Lemma
- Mean value inequality for a differentiable Banach-valued curve Lemma
- Neumann series and small perturbations of bounded inverses Lemma
- Point stabilizers, vertex residues, and rank-two boundary words Lemma
- The closed disk D² is a connected Hausdorff topological 2-manifold with boundary Lemma
- The interior-disc and closed-disc configuration spaces are homotopy equivalent Lemma
- A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm Theorem
- Bounded linear maps extend uniquely across the completion Theorem
- For a bilinear map, boundedness is equivalent to joint continuity Theorem
- Ordered configuration spaces cover the unordered ones regularly with deck group Sₙ Theorem
- The configuration braid short exact sequence 1→ PBₙ→ Bₙᶜᵒⁿᶠ→ Sₙ→ 1 Theorem
- The metric completion of a normed space carries a unique compatible Banach-space structure Theorem
- The norm completion of an inner-product space is a Hilbert space Theorem
Dependency tree · two levels
32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis (standard reference, not scraped)