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 ${\ displaystyle X}$ ${\ displaystyle U \ subset X}$ ${\ displaystyle X / U}$

{\ displaystyle \ | x + U \ |: = \ inf \ {\ | xy \ |; \, y \ in U \} = \ mathrm {dist} (x, U)}

.

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 . ${\ displaystyle U \ subset X}$ ${\ displaystyle X}$ ${\ displaystyle T: X \ rightarrow X / U}$ ${\ displaystyle X}$ ${\ displaystyle X / U}$ ${\ displaystyle U = \ mathrm {ker} (T)}$ ${\ displaystyle 1}$ ${\ displaystyle U}$ ${\ displaystyle 0}$

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 . ${\ displaystyle X, Y}$ ${\ displaystyle T: X \ rightarrow Y}$ ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle T}$ ${\ displaystyle X / \ mathrm {ker} (T) \ cong Y}$

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. ${\ displaystyle X}$ ${\ displaystyle U \ subset X}$ ${\ displaystyle X / U}$

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 . ${\ displaystyle X}$ ${\ displaystyle U \ subset X}$ ${\ displaystyle X / U}$

If there is a uniformly convex space and a closed subspace, then uniformly convex is also . ${\ displaystyle X}$ ${\ displaystyle U \ subset X}$ ${\ displaystyle X / U}$

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. ${\ displaystyle X}$ ${\ displaystyle U \ subset X}$ ${\ displaystyle X / U}$

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. ${\ displaystyle X}$ ${\ displaystyle U \ subset X}$ ${\ displaystyle X / U}$

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 ${\ displaystyle X}$ ${\ displaystyle {\ mathcal {P}}}$ ${\ displaystyle U \ subset X}$ ${\ displaystyle p \ in {\ mathcal {P}}}$ ${\ displaystyle {\ hat {p}}}$ ${\ displaystyle X / U}$

{\ displaystyle {\ hat {p}} (x + U): = \ inf \ {p (x + y); \, y \ in U \}}

.

Then the final topology agrees with the topology generated by the semi-norms , in particular the quotient space is again locally convex. ${\ displaystyle X / U}$ ${\ displaystyle \ {{\ hat {p}}; p \ in {\ mathcal {P}} \}}$

source

Dirk Werner : Functional Analysis . 6th, corrected edition, Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 , page 54