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.^[1] 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

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

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 ^[4]

[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

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

References

↑ Lumer, G. (1961), "Semi-inner-product spaces", Transactions of the American Mathematical Society, 100: 29–43, doi:10.2307/1993352, MR 0133024 .
↑ J. R. Giles, Classes of semi-inner-product spaces, Transactions of the American Mathematical Society 129 (1967), 436–446.
↑ J. B. Conway. A Course in Functional Analysis. 2nd Edition, Springer-Verlag, New York, 1990, page 1.
↑ S. V. Phadke and N. K. Thakare, When an s.i.p. space is a Hilbert space?, The Mathematics Student 42 (1974), 193–194.
↑ S. Dragomir, Semi-inner Products and Applications, Nova Science Publishers, Hauppauge, New York, 2004.
↑ 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.
↑ E. Torrance, Strictly convex spaces via semi-inner-product space orthogonality, Proceedings of the American Mathematical Society 26 (1970), 108–110.
↑ R. Der and D. Lee, Large-margin classification in Banach spaces, JMLR Workshop and Conference Proceedings 2: AISTATS (2007), 91–98.
↑ Haizhang Zhang, Yuesheng Xu and Jun Zhang, Reproducing kernel Banach spaces for machine learning, Journal of Machine Learning Research 10 (2009), 2741–2775.
↑ 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.

This article is issued from Wikipedia - version of the 12/5/2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.