# 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. Fundamental properties were later explored by Giles.

## Definition

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 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. $[f,f]\geq 0,$
5. $\left|[f,g]\right|\leq [f,f]^{1/2}[g,g]^{1/2}\quad \forall f,g\in V.$

## Difference from inner products

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

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

generally. This is equivalent to saying that 

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

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 a (not necessarily unique) 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 ($1\leq p<+\infty$ )
$\|x\|_{p}:={\biggl (}\sum _{j=1}^{n}|x_{j}|^{p}{\biggr )}^{1/p}$

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

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 $1\leq p<+\infty$ , with the norm
$\|f\|_{p}:=\left(\int _{\Omega }|f(t)|^{p}d\mu (t)\right)^{1/p}$

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.
2. In 2007, Der and Lee applied semi-inner-products to develop large margin classification in Banach spaces.
3. Recently, semi-inner-products have been used as the main tool in establishing the concept of reproducing kernel Banach spaces for machine learning.
4. Semi-inner-products can also be used to establish the theory of frames, Riesz bases for Banach spaces.