Quotient norm
A quotient standard or quotient semi-norm is in the functional analysis a generated naturally standard or semi-standard on a factor space .
definition
Let there be a normalized space and a sub-vector space . Define on the factor space
- .
Then this definition gives a semi-norm on the factor space; it is a norm precisely when the subspace is closed ; it is called the quotient norm or quotient semi-norm.
Quotient according to a kernel
If a closed subspace of the normalized space is , then the quotient mapping is linear, continuous, maps the open unit sphere from to the open unit sphere from and it is . The operator norm of the quotient mapping is , if there is a real subspace, otherwise the same .
Conversely, let there be normalized spaces and a linear mapping that maps the open unit sphere of to the open unit sphere of . Then is continuous , surjective and the isomorphism is an isometry .
properties
Many properties are inherited from the quotient norm:
- If a Banach space and a closed subspace is also a Banach space, i.e. H. the completeness inherited by the quotient norm.
- If a Hilbert space and a closed subspace is also a Hilbert space, i. H. the quotient norm is also generated by a scalar product .
- If there is a uniformly convex space and a closed subspace, then uniformly convex is also .
- If a Banach algebra and a closed two-sided ideal , then a Banach algebra, i.e. H. the submultiplicativity of the norm is carried over to the quotient norm.
- If a C * -algebra and a closed two-sided ideal, then a C * -algebra, i.e. H. the C * property of the norm also applies to the quotient norm.
Quotient semi-norms
The topology of a locally convex space is generated by a set of semi-norms. Be a subspace. For each , the quotient semi-norm is a semi-norm on the quotient space , where
- .
Then the final topology agrees with the topology generated by the semi-norms , in particular the quotient space is again locally convex.
source
- Dirk Werner : Functional Analysis . 6th, corrected edition, Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 , page 54