L-semi-inner product

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

1. ${\displaystyle [f+g,h]=[f,h]+[g,h]\quad \forall f,g,h\in V}$,
2. ${\displaystyle [\alpha f,g]=\alpha [f,g]\quad \forall \alpha \in \mathbb {C} ,\ \forall f,g\in V,}$
3. ${\displaystyle [f,\alpha g]={\overline {\alpha }}[f,g]\quad \forall \alpha \in \mathbb {C} ,\ \forall f,g\in V,}$
4. ${\displaystyle [f,f]\geq 0,}$
5. ${\displaystyle \left|[f,g]\right|\leq [f,f]^{1/2}[g,g]^{1/2}\quad \forall f,g\in V.}$

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

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

generally. This is equivalent to saying that [4]

${\displaystyle [f,g+h]\neq [f,g]+[f,h].\,}$

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

• If ${\displaystyle [\cdot ,\cdot ]}$ is a semi-inner-product for a linear vector space ${\displaystyle V}$ then

${\displaystyle \|f\|:=[f,f]^{1/2},\quad f\in V}$

defines a norm on ${\displaystyle V}$.

• Conversely, if ${\displaystyle V}$ is a normed vector space with the norm ${\displaystyle \|\cdot \|}$ then there always exists a (not necessarily unique) semi-inner-product on ${\displaystyle V}$ that is consistent with the norm on ${\displaystyle V}$ in the sense that

${\displaystyle \|f\|=[f,f]^{1/2},\ \ \forall f\in V.}$

• The Euclidean space ${\displaystyle \mathbb {C} ^{n}}$ with the ${\displaystyle \ell ^{p}}$ norm (${\displaystyle 1\leq p<+\infty }$)

${\displaystyle \|x\|_{p}:={\biggl (}\sum _{j=1}^{n}|x_{j}|^{p}{\biggr )}^{1/p}}$

has the consistent semi-inner-product:

${\displaystyle [x,y]:={\frac {\sum _{j=1}^{n}x_{j}{\overline {y_{j}}}|y_{j}|^{p-2}}{\|y\|_{p}^{p-2}}},\quad x,y\in \mathbb {C} ^{n}\setminus \{0\},\ \ 1<p<+\infty ,}$

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

where

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

• In general, the space ${\displaystyle L^{p}(\Omega ,d\mu )}$ of ${\displaystyle p}$-integrable functions on a measure space ${\displaystyle (\Omega ,\mu )}$, where ${\displaystyle 1\leq p<+\infty }$, with the norm

${\displaystyle \|f\|_{p}:=\left(\int _{\Omega }|f(t)|^{p}d\mu (t)\right)^{1/p}}$

possesses the consistent semi-inner-product:

${\displaystyle [f,g]:={\frac {\int _{\Omega }f(t){\overline {g(t)}}|g(t)|^{p-2}d\mu (t)}{\|g\|_{p}^{p-2}}},\ \ f,g\in L^{p}(\Omega ,d\mu )\setminus \{0\},\ \ 1<p<+\infty ,}$

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

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]