Let $$\mathcal {H}$$ be a Hilbert space, A positive definite operator in and $$\langle f,g\rangle _A=\langle Af,g\rangle $$ , $$f,g\in \mathcal the A-inner product. This paper studies geometry of set $$\begin{aligned} {I}_A^a:=\{\text { adjointable isometries for } \langle \ \rangle _A\}. \end{aligned}$$ It is proved that {I}_A^a$$ submanifold Banach algebra operators, homogeneous space group i...

Real inner product. Let V be a vector space over R. A (real) inner product is a function 〈−,−〉 : V × V → R such that • 〈x, y〉 = 〈y, x〉 for all x, y ∈ V, • 〈c1x1 + c2x2, y〉 = c1〈x1, y〉+ c2〈x2, y〉 for all x1, x2, y ∈ V, c1, c2 ∈ R, • 〈x, x〉 ≥ 0 with 〈x, x〉 = 0 iff x = 0. That is, the pairing is symmetric, linear in the first variable (and therefore bilinear, by symmetry), and positive definite. E...

We formalize ideas of orthogonality and inner products implicit in the development of a number of figures of merit (FOM) of color recording filters. We show that, in negligible measurement noise, the data dependence of each FOM based on linear color correction is equivalent to a choice of inner product (and hence of orthogonality). Further, we show that optimal sensors with respect to noise sen...

In this paper, we will define the inner-product and the norm in taxicab geometry and then we will discuss this inner-product geometrically.