# Semi-inner-product

A semi-inner-product is a generalization of inner products. It was introduced to mathematics by Lumer [1] for the purpose of extending Hilbert space type arguments to Banach spaces in functional analysis. Fundamental properties were later explored by Giles.[2]

## Definition

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 $V$ over the field $\mathbb{C}$ of complex numbers is a function from $V\times V$ to $\mathbb{C}$, usually denoted by $[\cdot,\cdot]$, such that

1. $[f+g,h]=[f,h]+[g,h]\quad \forall f,g,h\in V$,
2. $[\alpha f,g]=\alpha[f,g]\quad \forall \alpha\in\mathbb{C},\ \forall f,g\in V,$
3. $[f,\alpha g]=\overline{\alpha}[f,g] \quad \forall \alpha\in\mathbb{C},\ \forall f,g\in V,$
4. $[f,f]\ge 0\text{ and }[f,f]=0\text{ if and only if }f=0,$
5. $\left|[f,g]\right|\le [f,f]^{1/2}[g,g]^{1/2}\quad \forall f,g\in V.$

## Difference from inner products

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

$[f,g]\ne \overline{[g,f]}$

generally. This is equivalent to saying that [4]

$[f,g+h]\ne [f,g]+[f,h]. \,$

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

## Semi-inner-products for Banach spaces

• If $[\cdot,\cdot]$ is a semi-inner-product for a linear vector space $V$ then
$\|f\|:=[f,f]^{1/2},\quad f\in V$

defines a norm on $V$.

• Conversely, if $V$ is a normed vector space with the norm $\|\cdot\|$ then there always exists (maynot be unique) a semi-inner-product on $V$ that is consistent with the norm on $V$ in the sense that
$\|f\|=[f,f]^{1/2},\ \ \forall f\in V.$

## Examples

• The Euclidean space $\mathbb{C}^n$ with the $\ell^p$ norm ($1\le p<+\infty$)
$\|x\|_p:=\biggl(\sum_{j=1}^p |x_j|^p\biggr)^{1/p}$

has the consistent semi-inner-product:

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

where

$\operatorname{sgn}(t):=\left\{ \begin{array}{ll} \frac{t}{|t|},&t\in \mathbb{C}\setminus\{0\},\\ 0,&t=0. \end{array} \right.$
• In general, the space $L^p(\Omega,d\mu)$ of $p$-integrable functions on a measure space $(\Omega,\mu)$, where $1\le p<+\infty$, with the norm
$\|f\|_p:=\left(\int_\Omega |f(t)|^pd\mu(t)\right)^{1/p}$

possesses the consistent semi-inner-product:

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

## Applications

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

## References

1. ^ G. Lumer, Semi-inner-product spaces, Transactions of the American Mathematical Society 100 (1961), 29–43.
2. ^ J. R. Giles, Classes of semi-inner-product spaces, Transactions of the American Mathematical Society 129 (1967), 436–446.
3. ^ J. B. Conway. A Course in Functional Analysis. 2nd Edition, Springer-Verlag, New York, 1990, page 1.
4. ^ S. V. Phadke and N. K. Thakare, When an s.i.p. space is a Hilbert space?, The Mathematics Student 42 (1974), 193–194.
5. ^ S. Dragomir, Semi-inner Products and Applications, Nova Science Publishers, Hauppauge, New York, 2004.
6. ^ 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.
7. ^ E. Torrance, Strictly convex spaces via semi-inner-product space orthogonality, Proceedings of the American Mathematical Society 26 (1970), 108–110.
8. ^ R. Der and D. Lee, Large-margin classification in Banach spaces, JMLR Workshop and Conference Proceedings 2: AISTATS (2007), 91–98.
9. ^ Haizhang Zhang, Yuesheng Xu and Jun Zhang, Reproducing kernel Banach spaces for machine learning, Journal of Machine Learning Research 10 (2009), 2741–2775.
10. ^ 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.