Examples of "seminorm"
Similarly, a seminorm induces a pseudometric (see below), and a homogeneous, translation invariant pseudometric induces a seminorm.
Definition 1: A nuclear space is a locally convex topological vector space such that for any seminorm "p" we can find a larger seminorm "q" so that the natural map from "V" to "V" is nuclear.
Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder condition to the "L"-setting. For formula_55 and formula_56 the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by
A strongly nuclear space is a locally convex topological vector space such that for any seminorm "p" we can find a larger seminorm "q" so that the natural map from "V" to "V" is a strongly nuclear.
If "p" is a seminorm on "V", we write "V" for the Banach space given by completing "V" using the seminorm "p". There is a natural map from "V" to "V" (not necessarily injective).
We will say that a seminorm "p" is a Hilbert seminorm if "V" is a Hilbert space, or equivalently if "p" comes from a sesquilinear positive semidefinite form on "V".
It is the closure of the trigonometric polynomials under the seminorm
It is the closure of the trigonometric polynomials under the seminorm
A seminorm on "V" is a function with the properties 1. and 2. above.
More generally, for each real "p"≥1 we have the seminorm:
In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A sequence of vectors formula_32 is said to converge in norm to formula_33 if formula_34 as formula_35. Equivalently, the topology consists of all sets that can be represented as a union of open balls.
which is a seminorm on "X". The function "q" is a norm if and only if all "q" are norms.
be the standard deviation with respect to the seminorm . In this setting we can state the following:
is a seminorm on the vector space of all functions on which the Lebesgue integral on the right hand side is defined and finite. However, the seminorm is equal to zero for any function supported on a set of Lebesgue measure zero. These functions form a subspace which we "quotient out", making them equivalent to the zero function.
Definition 3: A nuclear space is a topological vector space with a topology defined by a family of Hilbert seminorms, such that for any Hilbert seminorm "p" we can find a larger Hilbert seminorm "q" so that the natural map from "V" to "V" is Hilbert–Schmidt.
Let be a set and let formula_29 be the space of all complex-valued functions on . Let be an increasing seminorm on formula_30 meaning that, for all real-valued functions formula_31 we have the following implication (the seminorm is also allowed to attain the value ∞):
The definition of many normed spaces (in particular, Banach spaces) involves a seminorm defined on a vector space and then the normed space is defined as the quotient space by the subspace of elements of seminorm zero. For instance, with the L spaces, the function defined by
Definition 2: A nuclear space is a topological vector space with a topology defined by a family of Hilbert seminorms, such that for any Hilbert seminorm "p" we can find a larger Hilbert seminorm "q" so that the natural map from "V" to "V" is trace class.
This is not to be confused with a seminorm or pseudonorm, where the norm axioms are satisfied except for positive definiteness.
Definition 4: A nuclear space is a locally convex topological vector space such that for any seminorm "p" the natural map from "V" to "V" is nuclear.