Calculator Inputs
Define a reference functional φ₀, a candidate functional φ, a finite test family, and the neighborhood tolerances. The calculator uses the weak-star neighborhood base generated by those test vectors.
Example Data Table
This sample shows a finite family that defines a neighborhood around a reference functional in a three-dimensional setting.
| Vector label | Coordinates | Reference φ₀(x) | Candidate φ(x) | Tolerance ε | Status |
|---|---|---|---|---|---|
| x1 | [1, 0, 2] | 5.0000 | 4.7000 | 0.4500 | Inside |
| x2 | [0, 1, -1] | -2.5000 | -2.2500 | 0.3000 | Inside |
| x3 | [2, -1, 1] | 4.5000 | 4.4500 | 0.6000 | Inside |
Formula Used
For a normed space X and dual space X*, a basic weak-star neighborhood around a reference functional φ₀ is generated by finitely many vectors x₁, …, xₘ and positive tolerances ε₁, …, εₘ:
U(φ₀; x₁, …, xₘ; ε₁, …, εₘ) = {φ ∈ X* : |φ(xᵢ) − φ₀(xᵢ)| < εᵢ for all i}.
In this calculator, functionals are represented by coefficient vectors over ℝⁿ, so each evaluation is a dot product:
φ(x) = Σ aⱼxⱼ and φ₀(x) = Σ bⱼxⱼ.
The neighborhood test is therefore the componentwise inequality |φ(xᵢ) − φ₀(xᵢ)| ≤ εᵢ for each supplied vector. The tool also computes a dual norm gap ‖φ − φ₀‖q, where q is dual to the chosen p-norm on X. Using Hölder’s inequality, each deviation satisfies:
|(φ − φ₀)(xᵢ)| ≤ ‖φ − φ₀‖q · ‖xᵢ‖p.
This gives a useful upper estimate that complements the direct neighborhood check.
How to Use This Calculator
- Choose the dimension n of the finite-dimensional model you want to study.
- Select the p-norm that describes the underlying geometry of the primal space.
- Enter the coefficients of the reference functional φ₀ and the candidate functional φ.
- Provide one test vector per line, each containing exactly n coordinates.
- Enter one tolerance for each vector, or a single tolerance applied to all vectors.
- Optionally enter a dual norm bound to test whether the candidate remains bounded.
- Submit the form to see the result summary above the form.
- Export the computed table as CSV or PDF when you need documentation.
Frequently Asked Questions
1. What does this weak-star topology calculator measure?
It checks whether a candidate functional stays inside a basic weak-star neighborhood around a reference functional, using finitely many test vectors and tolerance thresholds.
2. Why are only finitely many vectors used?
Basic weak-star neighborhoods are generated by finite constraint families. That makes the abstract topology computable while still matching the local neighborhood definition.
3. What are the coefficients of a functional here?
In the finite-dimensional model, a linear functional is represented by coefficients. The calculator evaluates it on each test vector through a dot product.
4. What does the slack margin show?
Slack equals tolerance minus actual deviation. Positive slack means the vector constraint is satisfied, while negative slack shows exactly how far the candidate misses the neighborhood condition.
5. Why does the calculator show a dual norm?
The dual norm measures boundedness of the functional relative to the chosen primal norm. It also supports the Hölder estimate for each evaluated deviation.
6. What is the Hölder bound used for?
It gives an upper estimate for each evaluation error. That helps compare observed deviations against a norm-based theoretical bound on the same test family.
7. Can one tolerance value be reused for all vectors?
Yes. If you enter a single tolerance, the calculator automatically applies it to every listed test vector before evaluating neighborhood membership.
8. Is this tool a full proof of weak-star convergence?
No. It analyzes a finite neighborhood test in a finite-dimensional model. That is useful for study, verification, and local approximation, not a general infinite-dimensional proof.