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equivalent norms
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(Definition)
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Let
and
be two norms on a vector space . These norms are equivalent norms if there exists a number such that
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(1) |
for all .
Since equation (1) is equivalent to
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(2) |
it follows that the definition is well defined. In other words,
and
are equivalent if and only if
and
are equivalent. An alternative condition is that there exist positive real numbers such that
However, this condition is equivalent to the above by setting
.
Some key results are as follows:
- If
and
, then
and
are equivalent. For example, if , then condition (1) holds with , and for , condition (2) holds with
.
- Suppose norms
and
are equivalent norms as in equation (1), and let and be the open balls with respect to
and
, respectively. By this result it follows that
It follows that the identity map from
to
is a homeomorphism. Or, alternatively, equivalent norms on induce the same topology on .
- The converse of the last paragraph is also true, i.e. if two norms induce the same topology on
then they are equivalent. This follows from the fact that every continuous linear function between two normed vector spaces is bounded (see this entry).
- Suppose
and
are inner product. Suppose further that the induced norms
and
are equivalent as in equation 1. Then, by the polarization identity, the inner products satisfy
- On a finite dimensional vector space all norms are equivalent (see this page). This is easy to understand as the unit sphere is compact if and only if a space is finite dimensional. On infinite dimensional spaces this result does not hold (see this page).
It follows that on a finite dimensional vector space, one can check continuity and convergence with respect with any norm. If a sequence converges in one norm, it converges in all norms. In matrix analysis this is particularly useful as one can choose the norm that is most easily calculated.
- The concept of equivalent norms also generalize to possibly non-symmetric norms. In this setting, all norms are also equivalent on a finite dimensional vector space. In particular,
and
are equivalent, and there exists such that
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"equivalent norms" is owned by matte. [ full author list (2) | owner history (2) ]
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Cross-references: matrix, converges, sequence, infinite dimensional, compact, unit sphere, finite dimensional, polarization identity, induced norms, inner product, normed vector spaces, function, continuous, converse, topology, induce, homeomorphism, identity map, open balls, real numbers, positive, well defined, equivalent, equation, number, vector space, norms
There are 2 references to this entry.
This is version 7 of equivalent norms, born on 2003-05-28, modified 2008-03-22.
Object id is 4312, canonical name is EquivalentNorms.
Accessed 10323 times total.
Classification:
| AMS MSC: | 46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous) |
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Pending Errata and Addenda
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