Let,, be the linear space of all -forms with summation over all ordered -tuples,.
Let be the linear space of all Lebesgue measurable functions, identifying the functions equal almost everywhere.
Let be the linear space of all with the property that.
(iii) Let be the linear space of all with the property that, for all and the set is finite. .
Let F ( [ 0, 1 ] τ, H ) be the linear space of the mesh functions φ τ = { φ k } 1 N − 1 with values in the Hilbert space H.
Let and let be the linear space of all Bochner measurable functions with the property that, which is a Banach space with respect to the norm (3.1).
Let w, l ∞, c and c 0 be the linear spaces of all, bounded, convergent and null sequences x = ( x k ) for all k ∈ N, respectively.
Let ℓ ∞, c and c 0 be the linear spaces of bounded, convergent and null sequences x = ( x k ) with complex terms, respectively, normed by ∥ x ∥ ∞ = sup k | x k |, where k ∈ N = { 1, 2, … }, the set of positive integers.
Let (s^{2}) denote the space of all double sequences, and let (ell_{infty }^{2}), (c^{2}) and (c_{0}^{2}) be the linear spaces of bounded, convergent and null sequences (x= ( x_{jk} ) ) with complex terms, respectively, normed by (Vert xVert _{ ( infty,2 ) }=sup_{j,k}vert x_{jk}vert ), where j, (kin mathbb{N}= { 1,2,, ldots } ).
Suppose that is the linear space of all -vectors, spanned by the exterior product corresponding to all ordered -tuples,.
where,,, and are twice continuously differentiable functions, is the linear space of all real symmetric matrices, and is the cone of all symmetric positive semidefinite matrices.
