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词汇 Continuous linear functional
分类 英语词汇 英语翻译词典
释义

Continuous linear functional

中文百科

线性泛函 Linear form

（重定向自Continuous linear functional）

在线性代数中，线性泛函是指由矢量空间到对应纯量域的线性映射。在 \mathbb{R}^n ，若矢量空间的矢量以列矢量表示；线性泛函则会以行矢量表示，在矢量上的作用则为它们的矩阵积。一般地，如果 V 是域 k 上的矢量空间，线性泛函 f 是一个从 V 到 k 的函数，它有以下的线性特性：

f(\vec{v}+\vec{w}) = f(\vec{v})+f(\vec{w}) \quad \forall \ \vec{v}, \vec{w}\in V
f(a\vec{v}) = af(\vec{v}) \quad \forall\ \vec{v}\in V, a\in k

所有从 V 到 k 的线性泛函集合, 记为 \operatorname{Hom}_k(V,k), 本身即为一矢量空间，称为 V 的 (代数)对偶空间。

英语百科

Linear form 线性泛函

（重定向自Continuous linear functional）
Geometric interpretation of a 1-form α as a stack of hyperplanes of constant value, each corresponding to those vectors that α maps to a given scalar value shown next to it along with the
Linear functionals (1-forms) α, β and their sum σ and vectors u, v, w, in 3d Euclidean space. The number of (1-form) hyperplanes intersected by a vector equals the inner product.[1]

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars. In ℝ, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the matrix product with the row vector on the left and the column vector on the right.  In general, if V is a vector space over a field k, then a linear functional f is a function from V to k that is linear:
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