Consider a lattice type g and let G be the corresponding space-group type.
Let γ be a Gaussian measure on a locally convex space and H be the corresponding Cameron Martin space.
Let μ:π↦μπ be a non-negative function defined on the dual space ˆG and let L2 be the corresponding Hilbert space which consists of elements π∈suppμ satisfying ∑μπTr<∞, where ξπ is a linear operator on the representation space of π, and is equipped with the inner product: ="∑μπTr.
We also let (H^{2}_{mathcal {X}} equiv H^{2}_{mathcal {X}}(mathbb {T})) be the corresponding Hardy space and (H^{infty}_{mathcal {X}} equiv H^{infty}_{mathcal {X}}(mathbb {T}) =L^{infty}_{mathcal {X}}cap H^{2}_{mathcal {X}}).
Let ((X,q)) be a quasi-partial metric space, let ((X,p_{q})) be the corresponding partial-metric space, and let ((X,d_{p_{q} })) be the corresponding metric space.
Definition 2.11 Let ( X, d ) be a metric space, and let ( X m, d ) be the corresponding product metric space.
A mapping (f:Xto X) is said to be continuous at (x_{0} in X) if, for every (varepsilon>0), there exists (delta>0) such that (f(B(x_{0},delta))subset B(f(x_{0}),varepsilon)). Let ((X,qp_{b})) be a quasi-partial b-metric space and ((X,d_{qp_{b} })) be the corresponding b-metric space.
Let (rho_{H}) be a monotone quasi-norm on (M^) and let H be the corresponding quasi-normed space, consisting of all locally integrable functions on ((0,1)) with a finite quasi-norm (|g|_{H}=rho_{H} |g|)).
Let (E, H, m) be an abstract Wiener space and (Ω, H, γ) be the corresponding Ito's Wiener space whereΩconsists of all the linear (but not necessarily continuous) functionals on the Hilbert spaceH.
Let (rho_{E}) be a monotone quasi-norm on (M^) and let E be the corresponding rearrangement invariant quasi-normed space consisting of all (fin L^{1}(Omega)) such that (|f|_{E}=rho_{E}(f^{ast})
Based on the basic tetragonal structure, the crystal lattice structures of the orthorhombic and monoclinic variants were determined and the corresponding space groups were analyzed according to the variation of the symmetrical elements.
