L-semi-inner product

Generalization of inner products that applies to all normed spaces

In mathematics, there are two different notions of semi-inner-product. The first, and more common, is that of an inner product which is not required to be strictly positive. This article will deal with the second, called a L-semi-inner product or semi-inner product in the sense of Lumer, which is an inner product not required to be conjugate symmetric. It was 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

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 {\displaystyle V} over the field C {\displaystyle \mathbb {C} } of complex numbers is a function from V × V {\displaystyle V\times V} to C , {\displaystyle \mathbb {C} ,} usually denoted by [ ⋅ , ⋅ ] {\displaystyle [\cdot ,\cdot ]} , such that for all f , g , h ∈ V : {\displaystyle f,g,h\in V:}

  1. Nonnegative-definiteness: [ f , f ] ≥ 0 , {\displaystyle [f,f]\geq 0,}
  2. Linearity in the 1st argument, meaning:
    1. Additivity in the 1st argument: [ f + g , h ] = [ f , h ] + [ g , h ] , {\displaystyle [f+g,h]=[f,h]+[g,h],}
    2. Homogeneity in the 1st argument: [ s f , g ] = s [ f , g ]  for all  s ∈ C , {\displaystyle [sf,g]=s[f,g]\quad {\text{ for all }}s\in \mathbb {C} ,}
  3. Conjugate homogeneity in the 2nd argument: [ f , s g ] = s ¯ [ f , g ]  for all  s ∈ C , {\displaystyle [f,sg]={\overline {s}}[f,g]\quad {\text{ for all }}s\in \mathbb {C} ,}
  4. Cauchy-Schwarz inequality: | [ f , g ] | ≤ [ f , f ] 1 / 2 [ g , g ] 1 / 2 . {\displaystyle |[f,g]|\leq [f,f]^{1/2}[g,g]^{1/2}.}

Difference from inner products

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

[ f , g ] ≠ [ g , f ] ¯ {\displaystyle [f,g]\neq {\overline {[g,f]}}}
generally. This is equivalent to saying that [4]
[ f , g + h ] ≠ [ f , g ] + [ f , h ] . {\displaystyle [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 normed spaces

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

‖ f ‖ := [ f , f ] 1 / 2 , f ∈ V {\displaystyle \|f\|:=[f,f]^{1/2},\quad f\in V}
defines a norm on V {\displaystyle V} .

Conversely, if V {\displaystyle V} is a normed vector space with the norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} then there always exists a (not necessarily unique) semi-inner-product on V {\displaystyle V} that is consistent with the norm on V {\displaystyle V} in the sense that[1]

‖ f ‖ = [ f , f ] 1 / 2 ,      for all  f ∈ V . {\displaystyle \|f\|=[f,f]^{1/2},\ \ {\text{ for all }}f\in V.}

Examples

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

‖ x ‖ p := ( ∑ j = 1 n | x j | p ) 1 / p {\displaystyle \|x\|_{p}:={\biggl (}\sum _{j=1}^{n}|x_{j}|^{p}{\biggr )}^{1/p}}
has the consistent semi-inner-product:
[ x , y ] := ∑ j = 1 n x j y j ¯ | y j | p − 2 ‖ y ‖ p p − 2 , x , y ∈ C n ∖ { 0 } ,     1 < p < + ∞ , {\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 ,}
[ x , y ] := ∑ j = 1 n x j sgn ⁡ ( y j ¯ ) , x , y ∈ C n ,     p = 1 , {\displaystyle [x,y]:=\sum _{j=1}^{n}x_{j}\operatorname {sgn} ({\overline {y_{j}}}),\quad x,y\in \mathbb {C} ^{n},\ \ p=1,}
where
sgn ⁡ ( t ) := { t | t | , t ∈ C ∖ { 0 } , 0 , t = 0. {\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 L p ( Ω , d μ ) {\displaystyle L^{p}(\Omega ,d\mu )} of p {\displaystyle p} -integrable functions on a measure space ( Ω , μ ) , {\displaystyle (\Omega ,\mu ),} where 1 ≤ p < + ∞ , {\displaystyle 1\leq p<+\infty ,} with the norm

‖ f ‖ p := ( ∫ Ω | f ( t ) | p d μ ( t ) ) 1 / p {\displaystyle \|f\|_{p}:=\left(\int _{\Omega }|f(t)|^{p}d\mu (t)\right)^{1/p}}
possesses the consistent semi-inner-product:
[ f , g ] := ∫ Ω f ( t ) g ( t ) ¯ | g ( t ) | p − 2 d μ ( t ) ‖ g ‖ p p − 2 ,     f , g ∈ L p ( Ω , d μ ) ∖ { 0 } ,     1 < p < + ∞ , {\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 ,}
[ f , g ] := ∫ Ω f ( t ) sgn ⁡ ( g ( t ) ¯ ) d μ ( t ) ,     f , g ∈ L 1 ( Ω , d μ ) . {\displaystyle [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.[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]

See also

  • Additive map – Z-module homomorphism
  • Cauchy's functional equation – Functional equation
  • Inner product – Generalization of the dot product; used to define Hilbert spacesPages displaying short descriptions of redirect targets

References

  1. ^ a b Lumer, G. (1961), "Semi-inner-product spaces", Transactions of the American Mathematical Society, 100: 29–43, doi:10.2307/1993352, MR 0133024.
  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.
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