7.  Properties in topological vector spaces

7.1 Fundamental concepts

The properties described in this section are partly algebraic, partly topological, and relate in particular to the notions of supplementary spaces, bases, convergent sequences and summable families. The notions indicated are a priori well known to engineers for finite-dimensional vector spaces, but cover many pitfalls when the dimension is infinite.

• Let E be a topological vector space, and M a vector subspace of E. A vector subspace N of E is said to be :

  • algebraic supplementary of M in E, if the application (x, y ) → x + y is an isomorphism of M × N on E for the vector space structure (i.e. a bijective linear application of M × N on E ) ;

  • ...