Encyclopedia > F-space

  Article Content


In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d : V × VR so that
  1. Scalar multiplication in V is continuous with respect to d and the standard metric on R or C.
  2. Addition in V is continuous with respect to d.
  3. The metric is translation-invariant, i.e. d(x+a, y+a) = d(x, y) for all x, y and a in V
  4. The metric space (V, d) is complete

Some authors call these spaces "Fréchet spaces", but in Wikipedia the term Fréchet space is reserved for locally convex[?] F-spaces.

Clearly, all Banach spaces and Fréchet spaces are F-spaces. The Lp spaces for 0 < p < 1 are examples of F-spaces which are not Fréchet spaces.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Quadratic formula

... of which are complex numbers. The two solutions are complex conjugates of each other. (In this case, the parabola does not intersect the x-axis at all.) Note that when ...

This page was created in 36.9 ms