英和和英

名詞

dual space (複数形 dual spaces)

1. (mathematics) The vector space which comprises the set of linear functionals of a given vector space.
2. (mathematics) The vector space which comprises the set of continuous linear functionals of a given topological vector space.
   • 2011, David E. Stewart, Dynamics with Inequalities, Society for Industrial and Applied Mathematics, page 17,

     The dual space of a Banach space ${\displaystyle X}$ is the vector space of continuous linear functions ${\displaystyle X\rightarrow \mathbb {R} }$, which are called functionals. Similar notation is used for duality pairing between the Banach space ${\displaystyle X}$ and its dual space ${\displaystyle X'}$: ${\displaystyle \left\langle u,v\right\rangle }$ is the result of applying the functional ${\displaystyle u\in X'}$ to ${\displaystyle v\in X}$: ${\displaystyle \left\langle u,v\right\rangle =u(v)}$ explicitly uses the fact that ${\displaystyle u}$ is a function ${\displaystyle X\rightarrow \mathbb {R} }$.