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← Revision 3 as of 2026-03-01 03:27:31 ⇥
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| * ''||a|| ≥ 0'' | * ''||a|| ≥ 0'' and ''||a|| = 0'' only if ''a'' is the zero vector |
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| * ''||a + b|| ≤ ||a|| + ||b||'' The third property is called the '''triangle inequality''', and it follows from the '''Cauchy-Schwartz inequality''' (i.e., ''|⟨u, v⟩|^2^ ≤ ⟨u, u⟩⟨v, v⟩'') which itself is a property of inner products. |
* The [[Analysis/CauchySchwarzInequality|triangle inequality]]: ''||a + b|| ≤ ||a|| + ||b||'' |
Norm
A norm is a generalized distance.
Contents
Description
In Euclidean spaces, the Euclidean distance describes the magnitude of a vector.
The idea of distance is generalized for inner product spaces as the norm. The natural norm for an inner product space is defined using the inner product: ||a|| = √⟨a, a⟩.
This is not the only feasible norm. A norm must satisfy these properties:
||a|| ≥ 0 and ||a|| = 0 only if a is the zero vector
||ca|| = |c| ||a|| for any scalar c
The triangle inequality: ||a + b|| ≤ ||a|| + ||b||
Other feasible norms include:
- The p-norm
If a = [x y z], then ||a||p = p√(|x|p + |y|p + |z|p).
Note that Euclidean distance is equivalent to the 2-norm. The absolute value operators are unnecessary for even ps.
- Max norm
If a = [x y z], then ||a||∞ = max{|x|, |y|, |z|}.