About Maximum Norm (L∞)
The Maximum norm, also known as L∞ norm, Chebyshev norm, or uniform norm, measures the largest
absolute value among a vector's components. It represents the maximum deviation from zero in
any coordinate direction.
Formula:
For a vector \(\vector{x} = (x_1, x_2, \ldots, x_n)\):
\[ \norm{\vector{x}}_\infty = \max_{1 \leq i \leq n} \abs{x_i} = \max(\abs{x_1}, \abs{x_2}, \ldots, \abs{x_n}) \]
Properties:
- Non-negativity: \(\norm{\vector{x}}_\infty \geq 0\)
- Positive definiteness: \(\norm{\vector{x}}_\infty = 0\) if and only if \(\vector{x} = \vector{0}\)
- Homogeneity: \(\norm{\alpha\vector{x}}_\infty = \abs{\alpha}\norm{\vector{x}}_\infty\)
- Triangle inequality: \(\norm{\vector{x} + \vector{y}}_\infty \leq \norm{\vector{x}}_\infty + \norm{\vector{y}}_\infty\)
- Limit of p-norms: \(\norm{\vector{x}}_\infty = \lim_{p \to \infty} \norm{\vector{x}}_p\)
Applications:
- Error bounds in numerical analysis
- Convergence analysis in iterative methods
- Minimax optimization problems
- Quality control and tolerance checking
- Digital signal processing
Example:
For the vector \(\vector{x} = (3, -5, 2, 4)\):
\[ \norm{\vector{x}}_\infty = \max(\abs{3}, \abs{-5}, \abs{2}, \abs{4}) = \max(3, 5, 2, 4) = 5 \]
Relationship to Other Norms:
For an \(n\)-dimensional vector \(\vector{x}\):
\[ \norm{\vector{x}}_\infty \leq \norm{\vector{x}}_1 \leq n\norm{\vector{x}}_\infty \]
\[ \norm{\vector{x}}_\infty \leq \norm{\vector{x}}_2 \leq \sqrt{n}\norm{\vector{x}}_\infty \]