Dual norm
From Infogalactic: the planetary knowledge core
The concept of a dual norm arises in functional analysis, a branch of mathematics.
Let be a normed space (or, in a special case, a Banach space) over a number field (i.e. or ) with norm . Then the dual (or conjugate) normed space (another notation ) is defined as the set of all continuous linear functionals from into the base field . If is such a linear functional, then the dual norm[1] of is defined by
With this norm, the dual space is also a normed space, and moreover a Banach space, since is always complete.[2]
Examples
- Dual Norm of Vectors
- If p, q ∈ satisfy , then the ℓp and ℓq norms are dual to each other.
- In particular the Euclidean norm is self-dual (p = q = 2). Similarly, the Schatten p-norm on matrices is dual to the Schatten q-norm.
- For , the dual norm is with positive definite.
- Dual Norm of Matrices
- Frobenius norm
- Its dual norm is
- Singular value norm
- Dual norm
- Frobenius norm
Notes
References
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