In
mathematics, there are two different notions of semi-inner-product. The first, and more common, is that of an inner product which is not required to be strictly positive. This article will deal with the second, called a L-semi-inner product or semi-inner product in the sense of Lumer, which is an inner product not required to be conjugate symmetric. It was formulated by
Günter Lumer
Günter Lumer (1929–2005) was a mathematician known for his work in functional analysis. He is the namesake of the Lumer–Phillips theorem on semigroups of operators on Banach spaces, and was the first to study L-semi-inner products. Born in G ...
, for the purpose of extending
Hilbert space type arguments to
Banach spaces
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
in
functional analysis. Fundamental properties were later explored by Giles.
Definition
We mention again that the definition presented here is different from that of the "semi-inner product" in standard functional analysis textbooks, where a "semi-inner product" satisfies all the properties of
inner products (including conjugate symmetry) except that it is not required to be strictly positive.
A semi-inner-product, L-semi-inner product, or a semi-inner product in the sense of Lumer for a
linear vector space over the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of complex numbers is a function from
to
usually denoted by