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Any norm on a one-dimensional vector space is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving isomorphism of vector spaces :, where is either or , and norm-preserving means that | | = (()).
Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.
All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology (although the resulting metric spaces need not be the same). [2] And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are Banach spaces.
The H p-quasinorm ||f || Hp of a distribution f of H p is defined to be the L p norm of M Φ f (this depends on the choice of Φ, but different choices of Schwartz functions Φ give equivalent norms). The H p-quasinorm is a norm when p ≥ 1, but not when p < 1. If 1 < p < ∞, the Hardy space H p is the same vector space as L p, with
The Frobenius norm defined by ‖ ‖ = = = | | = = = {,} is self-dual, i.e., its dual norm is ‖ ‖ ′ = ‖ ‖.. The spectral norm, a special case of the induced norm when =, is defined by the maximum singular values of a matrix, that is, ‖ ‖ = (), has the nuclear norm as its dual norm, which is defined by ‖ ‖ ′ = (), for any matrix where () denote the singular values ...
See this footnote for an example of a continuous norm on a Banach space that is not equivalent to that Banach space's given norm. [note 6] [11] All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space. [12] Complete norms vs complete metrics
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces .
Let Ω ⊂ R n be open and bounded, and let k ∈ N.Suppose that the Sobolev space H k (Ω) is compactly embedded in H k−1 (Ω). Then the following two norms on H k (Ω) are equivalent: