L-semi-inner product explained

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.

V over the field \Complex of complex numbers is a function from V x V to \Complex, usually denoted by [ ⋅ , ⋅ ] , such that for all f, g, h \in V:

1. Nonnegative-definiteness: [f,f] \geq 0 ,
2. Linearity in the 1st argument, meaning:
   1. Additivity in the 1st argument: [f + g,h] = [f,h] + [g,h],
   2. Homogeneity in the 1st argument: [s f,g] = s [f,g] forall s \in \Complex,
3. Conjugate homogeneity in the 2nd argument: [f, s g] = \overline{s} [f,g] forall s \in \Complex,
4. Cauchy-Schwarz inequality: |[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,$$[f,g]\; \backslash neq\; \backslash overline$$generally. This is equivalent to saying that ^[4] $$[f,g+h]\; \backslash neq\; [f,g]\; +\; [f,h].\; \backslash ,$$

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

Semi-inner-products for normed spaces

If

[ ⋅ , ⋅ ] is a semi-inner-product for a linear vector space V then $$\backslash |f\backslash |\; :=\; [f,f]^,\backslash quad\; f\backslash in\; V$$ defines a norm on V .

Conversely, if

V is a normed vector space with the norm \| ⋅ \| then there always exists a (not necessarily unique) semi-inner-product on V that is consistent with the norm on V in the sense that $$\backslash |f\backslash |\; =\; [f,f]^,\backslash \; \backslash \; \backslash text\; f\; \backslash in\; V.$$

Examples

\Complex ^{n} with the \ell ^{p} norm (1 \leq p< +inf ty ) $$\backslash |x\backslash |\_p\; :=\; \backslash biggl(\backslash sum\_^n\; |x\_j|^p\backslash biggr)^$$has the consistent semi-inner-product:

Notes and References

1. .
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.