L-semi-inner product

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, for the purpose of extending Hilbert space type arguments to Banach spaces in functional analysis.

[1] Fundamental properties were later explored by Giles.

[2] We mention again that the definition presented here is different from that of the "semi-inner product" in standard functional analysis textbooks,[3] 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

of complex numbers is a function from

A semi-inner-product is different from inner products in that it is in general not conjugate symmetric, that is,

In other words, semi-inner-products are generally nonlinear about its second variable.

is a semi-inner-product for a linear vector space

defines a norm on

is a normed vector space with the norm

then there always exists a (not necessarily unique) semi-inner-product on

that is consistent with the norm on

The Euclidean space

has the consistent semi-inner-product:

sgn ⁡ (

{\displaystyle [x,y]:=\|y\|_{1}\sum _{j=1}^{n}x_{j}\operatorname {sgn} ({\overline {y_{j}}}),\quad x,y\in \mathbb {C} ^{n},\ \ p=1,}

{\displaystyle \operatorname {sgn} (t):=\left\{{\begin{array}{ll}{\frac {t}{|t|}},&t\in \mathbb {C} \setminus \{0\},\\0,&t=0.\end{array}}\right.}

In general, the space

, d μ )

-integrable functions on a measure space

, μ ) ,

d μ ( t )

possesses the consistent semi-inner-product:

d μ ( t )

, d μ ) ∖ { 0 } ,

) d μ ( t ) ,

, d μ ) .