Semi-inner-product

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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[edit]

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[edit]

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[edit]

\|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[edit]

\|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}^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<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[edit]

  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[edit]

  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.