Factor space

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The quotient vector space , also called quotient space or factor space for short , is a term from linear algebra , a branch of mathematics . It is the vector space that arises as an image of a parallel projection along a sub-vector space . The elements of the quotient vector space are equivalence classes.


Let it be a vector space over a field and a subspace of . By fixing


is defined on an equivalence relation .

The vectors and are therefore equivalent if they differ by one vector . In other words: If the line through the points and to parallel are is, and equivalent.

The equivalence class of a point is


clearly shows the "parallel" affine subspace . The equivalence classes are also known as secondary classes (this term comes from group theory ).

The quotient vector space from to is the set of all equivalence classes and is denoted by:


It forms a vector space if the vector space operations are defined by proxy:

for and .

These operations are well-defined, i.e. independent of the choice of representatives.


  • Is a complement of in , i.e. H. is the direct sum of and , the restriction of to is an isomorphism . But there is no canonical possibility to understand it as a subspace of .
  • If it is finite-dimensional, then the following relationship results for the dimensions:
  • The dual space of can be identified with those linear forms on that are identical on.
  • The homomorphism theorem states that a linear mapping has an isomorphism
between the quotient space from after the kernel from and the image from induced, d. H. the concatenation
is the same .

Application in functional analysis

Many normed spaces created in the following way: Be a real or complex vector space and let 1 seminorm on . Then is a subspace of . The quotient space then becomes a normalized vector space with the norm .

More general: Let be a topological vector space that is not Hausdorffian . Then be analogous to above subspace define: . The quotient space with the quotient topology becomes a Hausdorff topological vector space.



The spaces and thus also the Sobolew spaces are quotient vector spaces.


The vector space and the one-dimensional sub-vector space are given . Then for example

an equivalence class of the quotient space .

Every straight line that is parallel to the bisecting line of the 1st quadrant is clearly an equivalence class:

Factor space.svg

See also