F-space
Sign in to saveIn functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex. Addition in X is continuous with respect to d. The metric is translation-invariant; that is, d(x + a, y + a) = d(x, y) for all x, y, a \in X. The metric space (X, d) is complete.
~3 min read
Article
9 sectionsContents
- Examples
- Example 1
- Example 2
- Sufficient conditions
- Related properties
- See also
- References
- Notes
- Sources
In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex. Addition in X is continuous with respect to d. The metric is translation-invariant; that is, d(x + a, y + a) = d(x, y) for all x, y, a \in X. The metric space (X, d) is complete.
The operation x \mapsto \|x\| := d(0, x) is called an F-norm, although in general an F-norm is not required to be homogeneous. By translation-invariance, the metric is recoverable from the F-norm. Thus, a real or complex F-space is equivalently a real or complex vector space equipped with a complete F-norm.