---- NB. This definition carries over to the discussion and verbiage of dual $R$-[[module|modules]], $R$ a [[commutative ring]], mostly cleanly. > [!definition] Definition. ([[dual vector space]]) > Let $V$ be [[vector space]] over [[field|field]] $\mathbb{F}$. We call [[vector space of linear maps between two vector spaces|the vector sapce]] $\hom(V, \ff)$ the **dual space** of $V$, and denote it $V^*$ or $V^{\vee}$. Its elements are variously called **linear functionals**, **linear forms**, **covectors**, **1-forms**, etc. > > Suppose further that $V$ is finite-dimensional. Upon fixing a [[basis]] $(\boldsymbol e_{1},\dots, \boldsymbol e_{n})$ of $V$, an element $\varphi \in V^{*}$ corresponds to a $1 \times n$ [[matrix]] (a **row vector**) written wrt the [[dual basis]] $(\varphi_{1},\dots, \varphi_{n})$. Under the [[isomorphism]] [[linear maps and basis of domain|determined by]] $\boldsymbol e_{i} \mapsto \varphi_{i}$, we have $\boldsymbol v = \sum_{i} v_{i} \boldsymbol e_{i} \mapsto \sum_{i} v_{i} \varphi_{i}$or as matrices ('in coordinates'): $\begin{bmatrix} v_{1} \\ \vdots \\ v_{n} \end{bmatrix} \mapsto \begin{bmatrix} v_{1} & \cdots & v_{n} \end{bmatrix}.$ Thus, $V \cong V^{*}$ by 'tipping vectors over'. *However*, this identification is not [[natural transformation|natural]]: it depended upon a choice of [[basis]]. > > What *is* [[natural transformation|natural]] is the [[double dual of a finite-dimensional vector space is naturally isomorphic to that space|identification]] $V \cong V^{**}$. The number $\varphi(v) \in \mathbb{F}$ obtained by applying a covector $\varphi$ to a vector $v$ is sometimes denoted by either of the more symmetric-looking notations $\langle \varphi, v \rangle$ or $\langle v, \varphi \rangle$. These expressions can be thought of as the action of $\varphi \in V^{*}$ on $v \in V$. Since $v$ is naturally identified with a [[linear functional]] $\xi(v):V^{*} \to \mathbb{F}$, $v \simeq \xi(v)$, they can also be thought of as the action of $\xi(v) \in V^{**}$ on $\varphi \in V^{*}$. Typically we omit mention of $\xi$ and think of $v$ either as a vector or as a [[linear functional]] $V^{*} \to \mathbb{F}$ depending on the context. > When $V$ is an [[dimension|infinite-dimensional]] [[normed vector space|normed vector space]], commonly one constrains $V^{*}$ to consist of *[[operator norm|bounded]]* linear functionals. (Some call this the **topological dual** of $V$.) - [ ] show the correspondence between bases and dual-dual bases (cf. LeeSM 275) > [!note] Remark. > Note that $\dim V = \dim V^*$ if $V$ is finite-dimensional. Hence $V \cong V^{*}$ per [[the rank theorem for free modules|the dimension theorem]]. ^note > [!justification] > From [[vector space of linear maps between two vector spaces#properties]] we have $\dim V^* = \dim V \cdot 1$, justifying the claim regarding [[dimension]]. ^justification ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```