This tool studies finite-dimensional vectors through norm-induced topology. It computes norms, distances, neighborhood membership, vector operations, scalar effects, and visual comparisons in one place.
Topological Vector Calculator
Plotly Graph
Two-dimensional inputs show geometric vectors. Higher-dimensional inputs show component comparisons.
Formula Used
Norms
- L1 norm: ||x||1 = Σ|xi|
- L2 norm: ||x||2 = √Σxi2
- L∞ norm: ||x||∞ = max|xi|
- Lp norm: ||x||p = (Σ|xi|p)1/p, for p > 0
Distance induced by the chosen norm
d(x, y) = ||x - y||
Scalar continuity identity
||αx|| = |α| ||x||
Open ball neighborhood
B(x, ε) = { y : ||y - x|| < ε }
This tool focuses on finite-dimensional real vectors. In finite dimensions, different norms may produce different numbers, yet they generate equivalent topologies.
How to Use This Calculator
- Enter the base vector
x. - Add vector
yif you want vector comparison. - Enter a test vector for neighborhood membership checks.
- Choose the norm that defines the topology.
- Enter
ponly for the Lp norm. - Set scalar
αto inspect scaling behavior. - Set radius
εfor the open ball test. - Press submit to generate results above the form.
- Use CSV or PDF buttons to export the results.
- Review the graph to interpret the vector geometry or components.
Example Data Table
| Example | x | y | α | ε | Norm | ||x|| | d(x, y) | Is y in B(x, ε)? |
|---|---|---|---|---|---|---|---|---|
| Sample 1 | (2, -1) | (1, 3) | 0.5 | 4 | L2 | 2.236 | 4.123 | No |
| Sample 2 | (3, 1, -2) | (2, 0, -1) | 2 | 3 | L1 | 6 | 3 | No |
| Sample 3 | (4, -2, 5) | (4, -1, 2) | -1 | 4 | L∞ | 5 | 3 | Yes |
Frequently Asked Questions
1. What does this tool calculate?
It calculates vector norms, induced distances, scalar transformations, vector addition, vector subtraction, and neighborhood membership using an open ball around the base vector.
2. What topology does the tool use?
It uses a norm-induced topology on finite-dimensional real vector spaces. Open balls generated by the selected norm provide the local neighborhoods used for continuity and closeness checks.
3. Why can I choose different norms?
Different norms measure size and distance differently. They help compare geometric behavior under L1, L2, L∞, and Lp settings, which is useful for analysis and teaching.
4. What is an open ball in this context?
An open ball B(x, ε) is the set of all vectors whose distance from x is smaller than ε under the selected norm. It represents a basic neighborhood around x.
5. Can this tool handle higher-dimensional vectors?
Yes. You can enter vectors with any matching finite dimension. For dimensions beyond two, the graph switches to a component comparison view instead of a geometric plane plot.
6. What does the scalar α show?
The scalar shows how vector scaling affects topology-related measurements. The tool verifies the identity ||αx|| = |α| ||x||, which reflects continuity of scalar multiplication.
7. Why do norms give different numeric answers?
Each norm emphasizes vector components differently. L1 adds absolute values, L2 measures Euclidean length, and L∞ tracks the largest component magnitude.
8. Is this tool suitable for formal proof work?
It is best for computation, intuition, and classroom demonstration. It does not replace a formal proof, but it helps verify examples and understand local topological behavior.