L-semi-inner product
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. Fundamental properties were later explored by Giles. Definition. We mention again that  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, L-semi-inner product, or a semi-inner product in the sense of Lumer for a linear vector space formula_1 over the field formula_2 of complex numbers is a function from formula_3 to formula_4 usually denoted by formula_5, such that for all formula_6 Difference from inner products. A semi-inner-product is different  from inner products in that it is in general not conjugate symmetric, that is, formula_7 generally. This is equivalent to saying that formula_8 In other words, semi-inner-products are generally nonlinear about its second variable. Semi-inner-products for normed spaces. If formula_5 is a semi-inner-product for a linear vector space formula_1 then formula_11 defines a norm on formula_1. Conversely, if formula_1 is a normed vector space with the norm formula_14 then there always exists a (not necessarily unique) semi-inner-product on formula_1 that is consistent with the norm on formula_16 in the sense that formula_17  Examples. The Euclidean space formula_18 with the formula_19 norm (formula_20) formula_21 has the consistent semi-inner-product: formula_22 where formula_23 In general, the space formula_24 of formula_25-integrable functions on a measure space formula_26 where formula_27 with the norm formula_28 possesses the consistent semi-inner-product: formula_29