- analysis, a
seminorm is like a norm but need not be
positive definite.
Seminorms are
intimately connected with
convex sets:
every seminorm is the Minkowski...
-
continuous seminorms on X {\displaystyle X} , then P {\displaystyle {\mathcal {P}}} is
called a base of
continuous seminorms if it is a base of
seminorms for...
-
Unlike seminorms, a
sublinear function does not have to be nonnegative-valued and also does not have to be
absolutely homogeneous.
Seminorms are themselves...
- is
complete with
respect to the
family of
seminorms. A
family P {\displaystyle {\mathcal {P}}} of
seminorms on X {\displaystyle X}
yields a Hausdorff...
- in particular,
every norm is also a
seminorm (and thus also a
sublinear functional). However,
there exist seminorms that are not norms.
Properties (1.)...
- of (continuous)
seminorms. The
topology of X {\displaystyle X} is
induced by a
countable increasing sequence of (continuous)
seminorms ( p i ) i = 1 ∞...
- {\displaystyle D.} Hahn–Banach
theorem for
seminorms — If p : M → R {\displaystyle p:M\to \mathbb {R} } is a
seminorm defined on a
vector subspace M {\displaystyle...
- all
seminorms p {\displaystyle p} ; it is
sufficient to
check it for a set of
seminorms that
generate the topology, in
other words, a set of
seminorms that...
-
seminormed spaces ( X i , q i ) {\displaystyle \left(X_{i},q_{i}\right)} with
seminorms q i : X i → R , {\displaystyle q_{i}:X_{i}\to \mathbb {R} ,}
denote the...
-
defined by the
seminorms indexed by x ∈ X: f ↦ ‖ f ( x ) ‖ Y . {\displaystyle f\mapsto \|f(x)\|_{Y}.} More generally, if a
family of
seminorms Q
defines the...
