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 ...
... • 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 ...
Geometry Chapter 5 Study Guide
... 1. In indirect proof, assume the ______________________to produce a ___________________. 2. What is the triangle inequality theorem? ...
... 1. In indirect proof, assume the ______________________to produce a ___________________. 2. What is the triangle inequality theorem? ...
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(+) ...
... 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(+) ...
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 ...
... 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 ...
1 Fields and vector spaces
... Writing scalars on the left, we have cd v c dv for all c d F and v V : that is, scalar multiplication by cd is the same as multiplication by d followed by multiplication by c, not vice versa. (The opposite convention would make V a right (rather than left) vector space; scalars would mor ...
... Writing scalars on the left, we have cd v c dv for all c d F and v V : that is, scalar multiplication by cd is the same as multiplication by d followed by multiplication by c, not vice versa. (The opposite convention would make V a right (rather than left) vector space; scalars would mor ...
V. Topological vector spaces
... V. Topological vector spaces V.1 Linear topologies and their generating Recalling of notation: R . . . the field of real numbers C . . . the field of complex numbers F . . . the field R or C If X is a vector space over F, the zero vector is denoted by o (and sometimes by 0). If X is a vector space o ...
... V. Topological vector spaces V.1 Linear topologies and their generating Recalling of notation: R . . . the field of real numbers C . . . the field of complex numbers F . . . the field R or C If X is a vector space over F, the zero vector is denoted by o (and sometimes by 0). If X is a vector space o ...
