# Semi-inner-product

In mathematics, the **semi-inner-product** is a generalization of inner products 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]}

## Contents

## DefinitionEdit

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** for a linear vector space over the field of complex numbers is a function from to , usually denoted by , such that

- ,

## Difference from inner productsEdit

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

generally. This is equivalent to saying that ^{[4]}

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

## Semi-inner-products for Banach spacesEdit

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

defines a norm on .

- Conversely, if 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 in the sense that

## ExamplesEdit

- The Euclidean space with the norm ( )

has the consistent semi-inner-product:

where

- In general, the space of -integrable functions on a measure space , where , with the norm

possesses the consistent semi-inner-product:

## ApplicationsEdit

- Following the idea of Lumer, semi-inner-products were widely applied to study bounded linear operators on Banach spaces.
^{[5]}^{[6]}^{[7]} - In 2007, Der and Lee applied semi-inner-products to develop large margin classification in Banach spaces.
^{[8]} - Recently, semi-inner-products have been used as the main tool in establishing the concept of reproducing kernel Banach spaces for machine learning.
^{[9]} - Semi-inner-products can also be used to establish the theory of frames, Riesz bases for Banach spaces.
^{[10]}

## ReferencesEdit

**^**Lumer, G. (1961), "Semi-inner-product spaces",*Transactions of the American Mathematical Society*,**100**: 29–43, doi:10.2307/1993352, MR 0133024.**^**J. R. Giles, Classes of semi-inner-product spaces, Transactions of the American Mathematical Society 129 (1967), 436–446.**^**J. B. Conway. A Course in Functional Analysis. 2nd Edition, Springer-Verlag, New York, 1990, page 1.**^**S. V. Phadke and N. K. Thakare, When an s.i.p. space is a Hilbert space?, The Mathematics Student 42 (1974), 193–194.**^**S. Dragomir, Semi-inner Products and Applications, Nova Science Publishers, Hauppauge, New York, 2004.**^**D. O. Koehler, A note on some operator theory in certain semi-inner-product spaces, Proceedings of the American Mathematical Society 30 (1971), 363–366.**^**E. Torrance, Strictly convex spaces via semi-inner-product space orthogonality, Proceedings of the American Mathematical Society 26 (1970), 108–110.**^**R. Der and D. Lee, Large-margin classification in Banach spaces, JMLR Workshop and Conference Proceedings 2: AISTATS (2007), 91–98.**^**Haizhang Zhang, Yuesheng Xu and Jun Zhang, Reproducing kernel Banach spaces for machine learning, Journal of Machine Learning Research 10 (2009), 2741–2775.**^**Haizhang Zhang and Jun Zhang, Frames, Riesz bases, and sampling expansions in Banach spaces via semi-inner products, Applied and Computational Harmonic Analysis 31 (1) (2011), 1–25.