Tikhonov's theorem (dynamical systems)

A semi-inner-product is a generalization of inner products. It was introduced to mathematics by 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. $[f,f]\geq 0{\text{ and }}[f,f]=0{\text{ if and only if }}f=0,$
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 (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 ($1\leq p<+\infty$)

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

has the consistent semi-inner-product:

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

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

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:

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

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

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.

References

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