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A '''semi-inner-product''' is a generalization of [[inner products]]. It was introduced to [[mathematics]] by Lumer <ref>G. Lumer, Semi-inner-product spaces, Transactions of the American Mathematical Society 100 (1961), 29–43.</ref> for the purpose of extending [[Hilbert space]] type arguments to [[Banach spaces]] in [[functional analysis]]. Fundamental properties were later explored by Giles.<ref>J. R. Giles, Classes of semi-inner-product spaces, Transactions of the American Mathematical Society 129 (1967), 436–446.</ref>

==Definition==
The definition presented here is different from that of the "semi-inner product" in standard functional analysis textbooks,<ref>J. B. Conway. A Course in Functional Analysis. 2nd Edition, Springer-Verlag, New York, 1990, page 1.</ref> 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]] <math>V</math> over the [[Number field|field]] <math>\mathbb{C}</math> of complex numbers is a function from <math>V\times V</math> to <math>\mathbb{C}</math>, usually denoted by <math>[\cdot,\cdot]</math>, such that
# <math>[f+g,h]=[f,h]+[g,h]\quad \forall f,g,h\in V</math>,
# <math>[\alpha f,g]=\alpha[f,g]\quad \forall \alpha\in\mathbb{C},\ \forall f,g\in V,</math>
# <math>[f,\alpha g]=\overline{\alpha}[f,g] \quad \forall \alpha\in\mathbb{C},\ \forall f,g\in V,</math>
# <math>[f,f]\ge 0\text{ and }[f,f]=0\text{ if and only if }f=0,</math>
#<math>\left|[f,g]\right|\le [f,f]^{1/2}[g,g]^{1/2}\quad \forall f,g\in V.</math>

==Difference from inner products==

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

::<math>[f,g]\ne \overline{[g,f]}</math>
generally. This is equivalent to saying that <ref>S. V. Phadke and N. K. Thakare, When an s.i.p. space is a Hilbert space?, The Mathematics Student 42 (1974), 193–194.</ref>

::<math>[f,g+h]\ne [f,g]+[f,h]. \, </math>

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

==Semi-inner-products for Banach spaces==
* If <math>[\cdot,\cdot]</math> is a semi-inner-product for a [[linear vector space]] <math>V</math> then

::<math>\|f\|:=[f,f]^{1/2},\quad f\in V</math>

defines a [[Norm (mathematics)|norm]] on <math>V</math>.

* Conversely, if <math>V</math> is a [[normed vector space]] with the [[Norm (mathematics)|norm]] <math>\|\cdot\|</math> then there always exists (maynot be unique) a semi-inner-product on <math>V</math> that is '''consistent''' with the norm on <math> V</math> in the sense that

::<math>\|f\|=[f,f]^{1/2},\ \ \forall f\in V.</math>

==Examples==
*The [[Euclidean space]] <math>\mathbb{C}^n</math> with the <math>\ell^p</math> norm (<math>1\le p<+\infty</math>)

::<math>\|x\|_p:=\biggl(\sum_{j=1}^p |x_j|^p\biggr)^{1/p}</math>

has the consistent semi-inner-product:

::<math>[x,y]:=\frac{\sum_{j=1}^n x_j\overline{y_j}|y_j|^{p-2}}{\|y\|_p^{p-2}},\quad x,y\in\mathbb{C}^n\setminus\{0\},\ \ 1<p<+\infty,</math>

::<math>[x,y]:=\sum_{j=1}^nx_j\operatorname{sgn}(\overline{y_j}),\quad x,y\in\mathbb{C}^n,\ \ p=1,</math>

where

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

*In general, the space <math>L^p(\Omega,d\mu)</math> of <math>p</math>-integrable functions on a [[measure space]] <math>(\Omega,\mu)</math>, where <math>1\le p<+\infty</math>, with the norm

::<math>\|f\|_p:=\left(\int_\Omega |f(t)|^pd\mu(t)\right)^{1/p}</math>

possesses the consistent semi-inner-product:

::<math>[f,g]:=\frac{\int_\Omega f(t)\overline{g(t)}|g(t)|^{p-2}d\mu(t)}{\|g\|_p^{p-2}},\ \ f,g\in L^p(\Omega,d\mu)\setminus\{0\},\ \ 1<p<+\infty,</math>

::<math>[f,g]:=\int_\Omega f(t)\operatorname{sgn}(\overline{g(t)})d\mu(t),\ \ f,g\in L^1(\Omega,d\mu).</math>

==Applications==
# Following the idea of Lumer, semi-inner-products were widely applied to study bounded linear operators on Banach spaces.<ref>S. Dragomir, Semi-inner Products and Applications, Nova Science Publishers, Hauppauge, New York, 2004.</ref><ref>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.</ref><ref>E. Torrance, Strictly convex spaces via semi-inner-product space orthogonality, Proceedings of the American Mathematical Society 26 (1970), 108–110.</ref>
# In 2007, Der and Lee applied semi-inner-products to develop large margin classification in Banach spaces.<ref>R. Der and D. Lee, Large-margin classification in Banach spaces, JMLR Workshop and Conference Proceedings 2: AISTATS (2007), 91–98.</ref>
# Recently, semi-inner-products have been used as the main tool in establishing the concept of reproducing kernel Banach spaces for machine learning.<ref>Haizhang Zhang, Yuesheng Xu and Jun Zhang, Reproducing kernel Banach spaces for machine learning, Journal of Machine Learning Research 10 (2009), 2741–2775.</ref>
# Semi-inner-products can also be used to establish the theory of frames, Riesz bases for Banach spaces.<ref>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.</ref>

==References==
{{Reflist}}

[[Category:Functional analysis]]