Linear and Bilinear Functionals
... Definition 1. A real linear functional is a mapping l (v) : V → R that is linear with respect to its argument v ∈ V.
That is, it must satisfy the properties
l (u + v) = l (u) + l (v)
l (αv) = αl (v)
for all u, v ∈ V and α ∈ R.
These are the same linearity properties used in the definition of a linea ...
linear vector space, V, informally. For a rigorous discuss
... A linear vector space, V, is a set of vectors with an abstract vector denoted by |vi (and read ‘ket vee’).
This notation introduced by Paul Adrien Maurice Dirac(1902-1984) is elegant and extremely useful
and it is imperative that you master it.1 The space is endowed with the operation of addition(+) ...
Geometry Chapter 5 Study Guide
... 1. In indirect proof, assume the ______________________to produce a ___________________.
2. What is the triangle inequality theorem?
3.2 Banach Spaces
... • Theorem 3.3.3 If S is not a vector subspace of E1 , then there is a unique
extension of L : S → E2 to a linear mapping L̃ : span S → E2 from the
vector subspace span S to E2 .
Proof: The extension L̃ is defined by linearity.
• Thus, one can always assume that the domain of a linear mapping is a ve ...
In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V together with a naturally induced linear structure. Dual vector spaces for finite-dimensional vector spaces show up in tensor analysis. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis.There are two types of dual spaces: the algebraic dual space, and the continuous dual space. The algebraic dual space is defined for all vector spaces. When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.