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. Fundamental properties were later explored by Giles.

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 for a linear vector space ${\displaystyle V}$ over the field ${\displaystyle \mathbb {C} }$ of complex numbers is a function from ${\displaystyle V\times V}$ to ${\displaystyle \mathbb {C} }$, usually denoted by ${\displaystyle [\cdot ,\cdot ]}$, such that

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