Semi-inner-product
The semi- inner product is a term from the mathematical branch of functional analysis. It is defined for vector spaces, where stands for the field of real or complex numbers, and generalizes the notion of inner product.
Definition
A semi- inner product on a vector space is a mapping with the following properties
Comparison with inner products
Is this a trivial way an inner product on the vector space, so satisfies the first two of the above conditions, and the Cauchy- Schwarz inequality shows that the third is met. Therefore, each inner product is a semi- inner product.
The converse is not true. What the semi- inner product is missing to be an inner product, the hermeticity and the linearity or Sesquilinearität in the second argument are.
Normed spaces
Is a semi- inner product on a vector space, so this is by defining a normed space. Conversely, one can show that every normed space is created in this way by a semi- inner product, that is, to each standard there is a semi- inner product, so that the above relation holds. That was the motivation for G. Lumer to introduce this concept. This has not to transfer the meaning as the inner product, but allowed in some situations, Hilbert space arguments far to Banach spaces.
A semi- product to a normalized inner space, that is, one which is represented by the above formula, the standard is not clear, in general. It can be shown that one can choose always such, which is homogeneous in the second argument, in other words, for this and is applicable to all. Here is the dash for the complex conjugation, which account in the case of real vector spaces.
Examples
- The standard Hilbert space on is given by a scalar product, generally, the standard is applied to an inner product space as the norm induced by the inner product.
- Lp- spaces: Is a measure space and is so set for:
Continuity properties
It is the set of all vectors of norm 1 of a normed vector space. A semi- inner product on a normed space is called continuous if for all, this is Re for the formation of the real part. In this term caution, because it does not mean that the semi- inner product is continuous as picture, above continuity property is obviously much weaker. It is said that the semi- inner product is uniformly continuous if the above limit equation is uniformly on the set.
This continuity properties can be brought with differentiability of the norm in conjunction. A normed space is called Gâteaux differentiable if
For all exists and uniformly Fréchet differentiable, if this limit exists uniformly on.
We have the following sentence:
- A semi- inner product is continuous (resp. uniformly continuous ) if the norm - Gâteaux differentiable (or uniformly Fréchet differentiable ) is.
The dual space
For a certain class of Banach spaces can be a prove to the representation theorem of Riesz - Frechét analogous set:
- Is a uniformly convex Banach space with a continuous semi- inner product, so there exists for every continuous linear functional on exactly one with all.
Course, this can not, as in the case of Hilbert spaces conclude that to its dual space is isomorphic, because the assignment in the above sentence is not linear in general. In the above example the rooms there is a uniformly convex Banach space with continuous semi- inner product. Every continuous linear functional has therefore the form with a. The emerging in the above integral is an item where. This is nothing more than the usual duality of rooms.
Numeric values range
The numerical range of a linear operator on a normed space can be described by an associated semi- inner product. The numerical range of is the conclusion of the convex hull of the set.