Alphabeta Math

R^n as a Normed Space; Vector-Valued Functions: Examples and Counterexamples

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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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✓ 10 results · all verified · 2 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full by a delegated reviewing agent on the owner's instruction; the judge is an additional, independent cross-model AI review of the proofs. The 8 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Counterexample

f(t) = (t², t³) on [0,1]: no ξ satisfies f(1)-f(0) = f'(ξ)

Counterexample

A curve for which the mean value inequality is an equality, showing the constant cannot be improved

Example

The comparison constants between ‖·‖₁, ‖·‖₂ and ‖·‖_∞ on ℝ², and vectors attaining each

Counterexample

‖·‖₁ on ℝ² violates the parallelogram law, so no symmetric bilinear form induces it

Counterexample

g(x,y) = xy/(x²+y²), extended by g(0,0)=0, is continuous in each variable separately and not continuous at the origin

Example

A convergent sequence in ℝ³ and the integral ∫₀¹ (1, t, t²), computed componentwise

Example

Steinitz's confinement bound realised on an explicit list of six unit vectors in ℝ² summing to zero

Example

A convergent series in ℝ² with Γ a line and Γ^⊥ a line, computed from the definition

False statement

FALSE: if a convergent series in ℝⁿ does not converge absolutely, then every point of ℝⁿ is the sum of some rearrangement of it

False statement

FALSE: all norms on a real vector space are equivalent

False statement

FALSE: in every normed space a closed bounded set is compact

False statement

FALSE: a sequence in ℝⁿ whose coordinate sequences are each bounded converges

Sources