In quantum mechanics, physical states are represented by rays in Hilbert space $\mathcal{H}$, which is a vector imbued an inner product $\ensuremath{\langle}\phantom{\rule{0.16em}{0ex}}|\phantom{\rule{0.16em}{0ex}}\ensuremath{\rangle}$, whose meaning arises as the overlap $\ensuremath{\langle}\ensuremath{\phi}|\ensuremath{\psi}\ensuremath{\rangle}$ for $|\ensuremath{\psi}\ensuremath{\rangle}$ p...

In this paper, for the first time, notion of multi complex numbers and number valued inner product is introduced in linear (vector) space. Starting from definition, some basic properties spaces are studied along with examples. Multi parallelogram law polarization identity established

Let X be a finite subset of Euclidean space R. We define for each x ∈ X, B(x) := {(x, y) | y ∈ X, x 6= y, (x, x) ≥ (y, y)} where (, ) denotes the standard inner product. X is called an inside s-inner product set if |B(x)| ≤ s for all x ∈ X. In this paper, we prove that the cardinalities of inside s-inner product sets have the Fisher type upper bound. An inside s-inner product set is said to be ...

with a symmetric and positive definite operator A defined on a Hilbert space V with dimV = N and its preconditioned version PCG. We derive the CG from the projection in A-inner product point of view. We shall use (·, ·) for the standard inner product and (·, ·)A for the inner product introduced by the SPD operator A. When we say ‘orthogonal’ vectors, we refer to the default (·, ·) inner product...