Vector norms: ||x||₁ = Σ|xᵢ| (Manhattan), ||x||₂ = √(Σxᵢ²) (Euclidean), ||x||∞ = max|xᵢ| (Chebyshev), ||x||p = (Σ|xᵢ|ᵖ)^(1/p). All satisfy: ||x|| ≥ 0, ||αx|| = |α|||x||, ||x+y|| ≤ ||x||+||y|| (triangle inequality).
How to Use
Enter a vector:
- Components: x₁, x₂, ...
- p: Norm parameter
- Output: All norms
Norm Comparison
||x||∞ ≤ ||x||₂ ≤ ||x||₁ ≤ n·||x||∞. All norms are equivalent in finite dimensions. L1 promotes sparsity, L2 is rotation-invariant, L∞ bounds worst-case.
In Machine Learning
- L1 regularization (Lasso): sparse solutions
- L2 regularization (Ridge): small solutions
- L∞: robust to outlier dimensions
Step-by-Step Instructions
- 1Enter vector components.
- 2View all norms.
- 3Compare values.
- 4Set custom p.
- 5Normalize vector.