Chapter 43: vector space

43.1 What is a vector?

What is a vector? or What is an element in a vector space?

Binary operations defined on a vector space satisfying some properties is more important than what is a vector.

ultimate answer: double dual concept^[43.4.1.2]

43.2 vector space definition

https://tex.stackexchange.com/a/141489 multiline node

Fig. 43.1: vector space construction

$\begin{aligned} & \begin{cases} F\text{ is a field} & \left(f\right)\text{field}\\ V\ne\emptyset & \left(ne\right)\text{nonempty set}\\ +:V\times V=V^{2}\overset{+}{\rightarrow}V\Leftrightarrow\forall\boldsymbol{u},\boldsymbol{v}\in V,\exists!\boldsymbol{w}\in V\left[\boldsymbol{w}=\boldsymbol{u}+\boldsymbol{v}\right] & \left(va\right)\text{vector addition}\\ \cdot:F\times V\overset{\cdot}{\rightarrow}V\Leftrightarrow\forall s\in F,\forall\boldsymbol{v}\in V,\exists!\boldsymbol{u}\in V\left[\boldsymbol{u}=s\boldsymbol{v}=s\cdot\boldsymbol{v}\right] & \left(sm\right)\text{scalar multiplication}\\ \begin{cases} \exists!\boldsymbol{0}\in V,\forall\boldsymbol{v}\in V\left[\boldsymbol{0}+\boldsymbol{v}=\boldsymbol{v}\right] & \left(e\right)\text{identity}\\ \forall\boldsymbol{v}\in V,\exists!-\boldsymbol{v}\in V\left[\left(-\boldsymbol{v}\right)+\boldsymbol{v}=\boldsymbol{0}\right] & \left(i\right)\text{inverse}\\ \forall\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}\in V\left[\boldsymbol{u}+\left(\boldsymbol{v}+\boldsymbol{w}\right)=\left(\boldsymbol{u}+\boldsymbol{v}\right)+\boldsymbol{w}\right] & \left(a\right)\text{associativity}\\ \forall\boldsymbol{u},\boldsymbol{v}\in V\left[\boldsymbol{u}+\boldsymbol{v}=\boldsymbol{v}+\boldsymbol{u}\right] & \left(c\right)\text{commutativity} \end{cases} & \left(va\right)\text{axioms}\\ \begin{cases} \exists!1\in F,\forall\boldsymbol{v}\in V\left[1\boldsymbol{v}=\boldsymbol{v}\right] & \left(e\right)\text{identity}\\ \forall s,t\in F,\boldsymbol{v}\in V\left[s\left(t\boldsymbol{v}\right)=\left(st\right)\boldsymbol{v}\right] & \left(a\right)\text{associativity}\\ \forall s,t\in F,\boldsymbol{v}\in V\left[\left(s+t\right)\boldsymbol{v}=s\boldsymbol{v}+t\boldsymbol{v}\right] & \left(ds\right)\text{scalar distributivity}\\ \forall s\in F,\boldsymbol{u},\boldsymbol{v}\in V\left[s\left(\boldsymbol{u}+\boldsymbol{v}\right)=s\boldsymbol{u}+s\boldsymbol{v}\right] & \left(dv\right)\text{vector distributivity} \end{cases} & \left(sm\right)\text{axioms} \end{cases}\\ \Leftrightarrow & V=V\left(F,+,\cdot\right)=\left(V,F,+,\cdot\right)\text{ is a vector space over the field }F\\ \Leftrightarrow & V\text{ is a vector space} \end{aligned}$

43.2.1 commutative group structure of vector space

$\left(va\right)$ axioms = vector addition axioms

$\begin{aligned} & V=\left(V,+\right)\text{ is a commutative group}\Leftrightarrow V=\left(V,+\right)\text{ is an abelian group}\\ \Leftrightarrow & \begin{cases} V=\left(V,+\right)=\left(V,+_{{\scriptscriptstyle V}}\right)\text{ is a group} & \left(g\right)\text{group}\\ \forall\boldsymbol{u},\boldsymbol{v}\in V\left[\boldsymbol{u}+\boldsymbol{v}=\boldsymbol{v}+\boldsymbol{u}\right] & \left(c\right)\text{commutativity} \end{cases}\\ \Leftrightarrow & \begin{cases} \begin{cases} +:V\times V=V^{2}\overset{+}{\rightarrow}V\Leftrightarrow\forall\boldsymbol{u},\boldsymbol{v}\in V,\exists!\boldsymbol{w}\in V\left[\boldsymbol{w}=\boldsymbol{u}+\boldsymbol{v}\right] & \left(cl\right)\text{closure}\\ \exists!\boldsymbol{0}\in V,\forall\boldsymbol{v}\in V\left[\boldsymbol{0}+\boldsymbol{v}=\boldsymbol{v}\right] & \left(e\right)\text{identity}\\ \forall\boldsymbol{v}\in V,\exists!-\boldsymbol{v}\in V\left[\left(-\boldsymbol{v}\right)+\boldsymbol{v}=\boldsymbol{0}\right] & \left(i\right)\text{inverse}\\ \forall\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}\in V\left[\boldsymbol{u}+\left(\boldsymbol{v}+\boldsymbol{w}\right)=\left(\boldsymbol{u}+\boldsymbol{v}\right)+\boldsymbol{w}\right] & \left(a\right)\text{associativity} \end{cases} & \left(g\right)\\ \forall\boldsymbol{u},\boldsymbol{v}\in V\left[\boldsymbol{u}+\boldsymbol{v}=\boldsymbol{v}+\boldsymbol{u}\right] & \left(c\right) \end{cases} \end{aligned}$

$\begin{aligned} &V\text{ is a vector space}\\\Leftrightarrow&V=V\left(F,+,\cdot\right)=\left(V,F,+,\cdot\right)\text{ is a vector space over the field }F\\\Leftrightarrow&\begin{cases} F\text{ is a field} & \left(f\right)\text{field}\\ V\ne\emptyset & \left(ne\right)\text{nonempty set}\\ V=\left(V,+\right)\text{ is a commutative group}\Leftrightarrow V=\left(V,+\right)\text{ is an abelian group} & \left(va\right)\text{vector addition}\\ \cdot:F\times V\overset{\cdot}{\rightarrow}V\Leftrightarrow\forall s\in F,\forall\boldsymbol{v}\in V,\exists!\boldsymbol{u}\in V\left[\boldsymbol{u}=s\boldsymbol{v}=s\cdot\boldsymbol{v}\right] & \left(sm\right)\text{scalar multiplication}\\ \begin{cases} \exists!1\in F,\forall\boldsymbol{v}\in V\left[1\boldsymbol{v}=\boldsymbol{v}\right] & \left(e\right)\text{identity}\\ \forall s,t\in F,\boldsymbol{v}\in V\left[s\left(t\boldsymbol{v}\right)=\left(st\right)\boldsymbol{v}\right] & \left(a\right)\text{associativity}\\ \forall s,t\in F,\boldsymbol{v}\in V\left[\left(s+t\right)\boldsymbol{v}=s\boldsymbol{v}+t\boldsymbol{v}\right] & \left(ds\right)\text{scalar distributivity}\\ \forall s\in F,\boldsymbol{u},\boldsymbol{v}\in V\left[s\left(\boldsymbol{u}+\boldsymbol{v}\right)=s\boldsymbol{u}+s\boldsymbol{v}\right] & \left(dv\right)\text{vector distributivity} \end{cases} & \left(sm\right)\text{axioms} \end{cases}\\\Leftrightarrow&\begin{cases} F\text{ is a field} & \left(f\right)\text{field}\\ V\ne\emptyset & \left(ne\right)\text{nonempty set}\\ \begin{cases} V=\left(V,+\right)=\left(V,+_{{\scriptscriptstyle V}}\right)\text{ is a group} & \left(g\right)\text{group}\\ \forall\boldsymbol{u},\boldsymbol{v}\in V\left[\boldsymbol{u}+\boldsymbol{v}=\boldsymbol{v}+\boldsymbol{u}\right] & \left(c\right)\text{commutativity} \end{cases} & \left(va\right)\text{vector addition}\\ \cdot=\cdot_{{\scriptscriptstyle F\times V}}:F\times V\overset{\cdot}{\rightarrow}V\Leftrightarrow\forall s\in F,\forall\boldsymbol{v}\in V,\exists!\boldsymbol{u}\in V\left[\boldsymbol{u}=s\boldsymbol{v}=s\cdot\boldsymbol{v}\right] & \left(sm\right)\text{scalar multiplication}\\ \begin{cases} \exists!1\in F,\forall\boldsymbol{v}\in V\left[1\boldsymbol{v}=\boldsymbol{v}\right] & \left(e\right)\text{identity}\\ \forall s,t\in F,\boldsymbol{v}\in V\left[s\left(t\boldsymbol{v}\right)=\left(st\right)\boldsymbol{v}\right] & \left(a\right)\text{associativity}\\ \forall s,t\in F,\boldsymbol{v}\in V\left[\left(s+t\right)\boldsymbol{v}=s\boldsymbol{v}+t\boldsymbol{v}\right] & \left(ds\right)\text{scalar distributivity}\\ \forall s\in F,\boldsymbol{u},\boldsymbol{v}\in V\left[s\left(\boldsymbol{u}+\boldsymbol{v}\right)=s\boldsymbol{u}+s\boldsymbol{v}\right] & \left(dv\right)\text{vector distributivity} \end{cases} & \left(sm\right)\text{axioms} \end{cases}\\\Leftrightarrow&\begin{cases} F=F\left(+_{{\scriptscriptstyle F}},\cdot_{{\scriptscriptstyle F}}\right)=\left(F,+_{{\scriptscriptstyle F}},\cdot_{{\scriptscriptstyle F}}\right)=\left(F,+,\cdot\right)\text{ is a field} & \left(f\right)\\ V\ne\emptyset & \left(ne\right)\\ \begin{cases} \begin{cases} +:V\times V=V^{2}\overset{+}{\rightarrow}V\Leftrightarrow\forall\boldsymbol{u},\boldsymbol{v}\in V,\exists!\boldsymbol{w}\in V\left[\boldsymbol{w}=\boldsymbol{u}+\boldsymbol{v}\right] & \left(cl\right)\text{closure}\\ \exists!\boldsymbol{0}\in V,\forall\boldsymbol{v}\in V\left[\boldsymbol{0}+\boldsymbol{v}=\boldsymbol{v}\right] & \left(e\right)\text{identity}\\ \forall\boldsymbol{v}\in V,\exists!-\boldsymbol{v}\in V\left[\left(-\boldsymbol{v}\right)+\boldsymbol{v}=\boldsymbol{0}\right] & \left(i\right)\text{inverse}\\ \forall\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}\in V\left[\boldsymbol{u}+\left(\boldsymbol{v}+\boldsymbol{w}\right)=\left(\boldsymbol{u}+\boldsymbol{v}\right)+\boldsymbol{w}\right] & \left(a\right)\text{associativity} \end{cases} & \left(g\right)\\ \forall\boldsymbol{u},\boldsymbol{v}\in V\left[\boldsymbol{u}+\boldsymbol{v}=\boldsymbol{v}+\boldsymbol{u}\right] & \left(c\right) \end{cases} & \left(va\right)\\ \begin{cases} \cdot:F\times V\overset{\cdot}{\rightarrow}V\Leftrightarrow\forall s\in F,\forall\boldsymbol{v}\in V,\exists!\boldsymbol{u}\in V\left[\boldsymbol{u}=s\boldsymbol{v}=s\cdot\boldsymbol{v}\right] & \left(cl\right)\text{closure}\\ \exists!1\in F,\forall\boldsymbol{v}\in V\left[1\boldsymbol{v}=\boldsymbol{v}\right] & \left(e\right)\text{identity}\\ \forall s,t\in F,\boldsymbol{v}\in V\left[s\left(t\boldsymbol{v}\right)=s\cdot_{{\scriptscriptstyle F\times V}}\left(t\cdot_{{\scriptscriptstyle F\times V}}\boldsymbol{v}\right)=\left(s\cdot_{{\scriptscriptstyle F}}t\right)\cdot_{{\scriptscriptstyle F\times V}}\boldsymbol{v}=\left(st\right)\boldsymbol{v}\right] & \left(a\right)\text{associativity}\\ \forall s,t\in F,\boldsymbol{v}\in V\left[\left(s+t\right)\boldsymbol{v}=\left(s+_{{\scriptscriptstyle F}}t\right)\boldsymbol{v}=s\boldsymbol{v}+_{{\scriptscriptstyle V}}t\boldsymbol{v}=s\boldsymbol{v}+t\boldsymbol{v}\right] & \left(ds\right)\text{scalar distributivity}\\ \forall s\in F,\boldsymbol{u},\boldsymbol{v}\in V\left[s\left(\boldsymbol{u}+\boldsymbol{v}\right)=s\boldsymbol{u}+s\boldsymbol{v}\right] & \left(dv\right)\text{vector distributivity} \end{cases} & \left(sm\right) \end{cases} \end{aligned}$

43.2.2 scalar distributivity

$\left(sm\right)\left(ds\right)$

$\forall s,t\in F,\boldsymbol{v}\in V\left[\left(s+t\right)\boldsymbol{v}=s\boldsymbol{v}+t\boldsymbol{v}\right]$

$\forall s,t\in F,\boldsymbol{v}\in V\left[\left(s+_{{\scriptscriptstyle F}}t\right)\boldsymbol{v}=s\boldsymbol{v}+t\boldsymbol{v}\right]$

$\forall s,t\in F,\boldsymbol{v}\in V\left[\left(s+_{{\scriptscriptstyle F}}t\right)\boldsymbol{v}=s\boldsymbol{v}+_{{\scriptscriptstyle V}}t\boldsymbol{v}\right]$

43.3 linearity

$\begin{aligned} & \begin{cases} f\left(x+y\right)=f\left(x\right)+f\left(y\right) & \text{additivity}\\ f\left(\lambda x\right)=\lambda f\left(x\right) & \text{homogeneity} \end{cases}\\ \Leftrightarrow & f\left(\lambda x+y\right)=\lambda f\left(x\right)+f\left(y\right)\\ \Leftrightarrow & f\text{ is linear} \end{aligned}$

43.3.1 linear structure of vector space

$\forall s\in F,\boldsymbol{u},\boldsymbol{v}\in V\left[\boldsymbol{u}+s\boldsymbol{v}\in V\right]$

$\forall s\in F,\forall\boldsymbol{u},\boldsymbol{v}\in V\left[\boldsymbol{u}+s\boldsymbol{v}\in V\right]$

$\forall s\in F,\left\langle \boldsymbol{u},\boldsymbol{v}\right\rangle \in V^{2}\left[\boldsymbol{u}+s\boldsymbol{v}\in V\right]$

$\begin{aligned} & \begin{cases} \forall\boldsymbol{u},\boldsymbol{v}\in V\left[\boldsymbol{u}+\boldsymbol{v}\in V\right] & \text{vector addition closure}\\ \forall s\in F,\boldsymbol{v}\in V\left[s\boldsymbol{v}\in V\right] & \text{scalar multiplication closure} \end{cases}\\ \Leftrightarrow & \begin{cases} \forall\boldsymbol{u},\boldsymbol{v}\in V\left[\boldsymbol{u}+\boldsymbol{v}\in V\right] & \left(a\right)\text{additivity}\\ \forall s\in F,\boldsymbol{v}\in V\left[s\boldsymbol{v}\in V\right] & \left(h\right)\text{homogeneity} \end{cases}\\ \Leftrightarrow & \forall s\in F,\boldsymbol{u},\boldsymbol{v}\in V\left[\boldsymbol{u}+s\boldsymbol{v}\in V\right]\text{ }\left(l\right)\text{linearity} \end{aligned}$

43.3.2 linear transformation or linear map

$\begin{aligned} & \begin{cases} V,W\text{ are vector spaces}\\ T:V\rightarrow W\\ \begin{cases} \forall\boldsymbol{u},\boldsymbol{v}\in V\left[T\left(\boldsymbol{u}+\boldsymbol{v}\right)=T\left(\boldsymbol{u}\right)+T\left(\boldsymbol{v}\right)\right] & \left(a\right)\text{additivity}\\ \forall\boldsymbol{v}\in V,c\in F\left[T\left(c\boldsymbol{v}\right)=cT\left(\boldsymbol{v}\right)\right] & \left(h\right)\text{homogeneity} \end{cases} & \left(L\right) \end{cases}\\ \Leftrightarrow & \begin{cases} V,W\text{ are vector spaces}\\ T:V\rightarrow W\\ \forall\boldsymbol{u},\boldsymbol{v}\in V,c\in F\left[T\left(\boldsymbol{u}+c\boldsymbol{v}\right)=T\left(\boldsymbol{u}\right)+cT\left(\boldsymbol{v}\right)\right] & \left(l\right)\text{linearity} \end{cases}\\ \Leftrightarrow & T\text{ is a linear map from }V\text{ to }W\\ \Leftrightarrow & T\text{ is a linear tranformation} \end{aligned}$

43.4 vector space example

arrow vector
number
- integer
- real
- complex
- quaternion
function
- polynomial function
- continuous function
matrix
- real matrix
- complex matrix
reciprocal space

https://www.bilibili.com/video/BV1NC4y1J7UL

applications in different disciplines

math
- recursive number series
- Fourier series
physics
- electrical circuit: linear response / superposition theorem in linear circuit / linear network
chemistry
- balancing chemical equation

43.4.1 reciprocal space

reciprocal space = 倒易空間

$\begin{cases} \boldsymbol{e}_{{\scriptscriptstyle 1}}=\boldsymbol{a} & \boldsymbol{a}\times\boldsymbol{b}\ne\boldsymbol{0}\\ \boldsymbol{e}_{{\scriptscriptstyle 2}}=\boldsymbol{b} & \boldsymbol{b}\times\boldsymbol{c}\ne\boldsymbol{0}\\ \boldsymbol{e}_{{\scriptscriptstyle 3}}=\boldsymbol{c} & \boldsymbol{c}\times\boldsymbol{a}\ne\boldsymbol{0} \end{cases}\Rightarrow\begin{cases} \boldsymbol{e}_{{\scriptscriptstyle 1}}^{\prime}=\dfrac{\boldsymbol{b}\times\boldsymbol{c}}{\Omega}\\ \boldsymbol{e}_{{\scriptscriptstyle 2}}^{\prime}=\dfrac{\boldsymbol{c}\times\boldsymbol{a}}{\Omega}\\ \boldsymbol{e}_{{\scriptscriptstyle 3}}^{\prime}=\dfrac{\boldsymbol{a}\times\boldsymbol{b}}{\Omega} \end{cases},$

$\Omega=\boldsymbol{a}\cdot\left(\boldsymbol{b}\times\boldsymbol{c}\right)=\boldsymbol{b}\cdot\left(\boldsymbol{c}\times\boldsymbol{a}\right)=\boldsymbol{c}\cdot\left(\boldsymbol{a}\times\boldsymbol{b}\right)$

reciprocal space as dual space and contravariant vector

$\begin{aligned} & \mathrm{span}\left\{ \boldsymbol{e}_{{\scriptscriptstyle 1}},\boldsymbol{e}_{{\scriptscriptstyle 2}},\boldsymbol{e}_{{\scriptscriptstyle 3}}\right\} =\mathrm{span}\left\{ \boldsymbol{a},\boldsymbol{b},\boldsymbol{c}\right\} =V\\ =\mathbb{R}^{3}= & \mathrm{span}\left\{ \boldsymbol{e}_{{\scriptscriptstyle 1}}^{\prime},\boldsymbol{e}_{{\scriptscriptstyle 2}}^{\prime},\boldsymbol{e}_{{\scriptscriptstyle 3}}^{\prime}\right\} =\mathrm{span}\left\{ \dfrac{\boldsymbol{b}\times\boldsymbol{c}}{\Omega},\dfrac{\boldsymbol{c}\times\boldsymbol{a}}{\Omega},\dfrac{\boldsymbol{a}\times\boldsymbol{b}}{\Omega}\right\} \\ = & \mathrm{span}\left\{ \boldsymbol{e}^{{\scriptscriptstyle 1}},\boldsymbol{e}^{{\scriptscriptstyle 2}},\boldsymbol{e}^{{\scriptscriptstyle 3}}\right\} =\mathrm{span}\left\{ \boldsymbol{e}^{{\scriptscriptstyle *}}\right\} _{{\scriptscriptstyle *\in\left\{ 1,2,3\right\} }}=V^{*} \end{aligned}$

43.4.1.1 Kronecker delta

$\begin{pmatrix}\boldsymbol{e}_{{\scriptscriptstyle 1}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 1}}^{\prime} & \boldsymbol{e}_{{\scriptscriptstyle 1}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 2}}^{\prime} & \boldsymbol{e}_{{\scriptscriptstyle 1}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 3}}^{\prime}\\ \boldsymbol{e}_{{\scriptscriptstyle 2}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 1}}^{\prime} & \boldsymbol{e}_{{\scriptscriptstyle 2}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 2}}^{\prime} & \boldsymbol{e}_{{\scriptscriptstyle 2}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 3}}^{\prime}\\ \boldsymbol{e}_{{\scriptscriptstyle 3}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 1}}^{\prime} & \boldsymbol{e}_{{\scriptscriptstyle 3}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 2}}^{\prime} & \boldsymbol{e}_{{\scriptscriptstyle 3}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 3}}^{\prime} \end{pmatrix}=\left[\delta_{{\scriptscriptstyle ij}}\right]=\begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}=\begin{pmatrix}\boldsymbol{e}^{{\scriptscriptstyle 1}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 1}} & \boldsymbol{e}^{{\scriptscriptstyle 1}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 2}} & \boldsymbol{e}^{{\scriptscriptstyle 1}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 3}}\\ \boldsymbol{e}^{{\scriptscriptstyle 2}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 1}} & \boldsymbol{e}^{{\scriptscriptstyle 2}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 2}} & \boldsymbol{e}^{{\scriptscriptstyle 2}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 3}}\\ \boldsymbol{e}^{{\scriptscriptstyle 3}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 1}} & \boldsymbol{e}^{{\scriptscriptstyle 3}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 2}} & \boldsymbol{e}^{{\scriptscriptstyle 3}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 3}} \end{pmatrix}$

Kronecker delta

$\boldsymbol{e}_{{\scriptscriptstyle i}}\cdot\boldsymbol{e}_{{\scriptscriptstyle j}}^{\prime}=\delta_{{\scriptscriptstyle ij}}=\begin{cases} 1 & i=j\\ 0 & i\ne j \end{cases}$

Kronecker delta tensor = Kronecker tensor

$\boldsymbol{e}^{{\scriptscriptstyle i}}\left(\boldsymbol{e}_{{\scriptscriptstyle j}}\right)=\boldsymbol{e}^{{\scriptscriptstyle i}}\cdot\boldsymbol{e}_{{\scriptscriptstyle j}}=\delta_{{\scriptscriptstyle j}}^{{\scriptscriptstyle i}}=\delta^{{\scriptscriptstyle i}}{}_{{\scriptscriptstyle j}}=\begin{cases} 1 & i=j\\ 0 & i\ne j \end{cases}$

$\boldsymbol{v}=v_{{\scriptscriptstyle a}}\boldsymbol{a}+v_{{\scriptscriptstyle b}}\boldsymbol{b}+v_{{\scriptscriptstyle c}}\boldsymbol{c}=v_{{\scriptscriptstyle 1}}\boldsymbol{e}_{{\scriptscriptstyle 1}}+v_{{\scriptscriptstyle 2}}\boldsymbol{e}_{{\scriptscriptstyle 2}}+v_{{\scriptscriptstyle 3}}\boldsymbol{e}_{{\scriptscriptstyle 3}}$

$\boldsymbol{e}^{{\scriptscriptstyle 1}}\cdot\boldsymbol{v}=\boldsymbol{v}\cdot\boldsymbol{e}_{{\scriptscriptstyle 1}}^{\prime}=v_{{\scriptscriptstyle 1}}\boldsymbol{e}_{{\scriptscriptstyle 1}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 1}}^{\prime}+v_{{\scriptscriptstyle 2}}\boldsymbol{e}_{{\scriptscriptstyle 2}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 1}}^{\prime}+v_{{\scriptscriptstyle 3}}\boldsymbol{e}_{{\scriptscriptstyle 3}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 1}}^{\prime}=v_{{\scriptscriptstyle 1}}$

$\boldsymbol{e}^{{\scriptscriptstyle 2}}\cdot\boldsymbol{v}=\boldsymbol{v}\cdot\boldsymbol{e}_{{\scriptscriptstyle 2}}^{\prime}=v_{{\scriptscriptstyle 1}}\boldsymbol{e}_{{\scriptscriptstyle 1}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 2}}^{\prime}+v_{{\scriptscriptstyle 2}}\boldsymbol{e}_{{\scriptscriptstyle 2}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 2}}^{\prime}+v_{{\scriptscriptstyle 3}}\boldsymbol{e}_{{\scriptscriptstyle 3}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 2}}^{\prime}=v_{{\scriptscriptstyle 2}}$

$\boldsymbol{e}^{{\scriptscriptstyle 3}}\cdot\boldsymbol{v}=\boldsymbol{v}\cdot\boldsymbol{e}_{{\scriptscriptstyle 3}}^{\prime}=v_{{\scriptscriptstyle 1}}\boldsymbol{e}_{{\scriptscriptstyle 1}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 3}}^{\prime}+v_{{\scriptscriptstyle 2}}\boldsymbol{e}_{{\scriptscriptstyle 2}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 3}}^{\prime}+v_{{\scriptscriptstyle 3}}\boldsymbol{e}_{{\scriptscriptstyle 3}}\cdot\boldsymbol{e}_{{\scriptscriptstyle 3}}^{\prime}=v_{{\scriptscriptstyle 3}}$

$\begin{aligned} \boldsymbol{v}= & v_{{\scriptscriptstyle 1}}\boldsymbol{e}_{{\scriptscriptstyle 1}}+v_{{\scriptscriptstyle 2}}\boldsymbol{e}_{{\scriptscriptstyle 2}}+v_{{\scriptscriptstyle 3}}\boldsymbol{e}_{{\scriptscriptstyle 3}}\\ = & \left(\boldsymbol{v}\cdot\boldsymbol{e}_{{\scriptscriptstyle 1}}^{\prime}\right)\boldsymbol{e}_{{\scriptscriptstyle 1}}+\left(\boldsymbol{v}\cdot\boldsymbol{e}_{{\scriptscriptstyle 2}}^{\prime}\right)\boldsymbol{e}_{{\scriptscriptstyle 2}}+\left(\boldsymbol{v}\cdot\boldsymbol{e}_{{\scriptscriptstyle 3}}^{\prime}\right)\boldsymbol{e}_{{\scriptscriptstyle 3}}\\ = & \left(\boldsymbol{e}^{{\scriptscriptstyle 1}}\cdot\boldsymbol{v}\right)\boldsymbol{e}_{{\scriptscriptstyle 1}}+\left(\boldsymbol{e}^{{\scriptscriptstyle 2}}\cdot\boldsymbol{v}\right)\boldsymbol{e}_{{\scriptscriptstyle 2}}+\left(\boldsymbol{e}^{{\scriptscriptstyle 3}}\cdot\boldsymbol{v}\right)\boldsymbol{e}_{{\scriptscriptstyle 3}}\\ = & \boldsymbol{e}^{{\scriptscriptstyle 1}}\left(\boldsymbol{v}\right)\boldsymbol{e}_{{\scriptscriptstyle 1}}+\boldsymbol{e}^{{\scriptscriptstyle 2}}\left(\boldsymbol{v}\right)\boldsymbol{e}_{{\scriptscriptstyle 2}}+\boldsymbol{e}^{{\scriptscriptstyle 3}}\left(\boldsymbol{v}\right)\boldsymbol{e}_{{\scriptscriptstyle 3}} \end{aligned}$

$\begin{cases} \boldsymbol{e}^{{\scriptscriptstyle 1}}\left(\boldsymbol{v}\right)=\boldsymbol{e}^{{\scriptscriptstyle 1}}\cdot\boldsymbol{v}=\boldsymbol{v}\cdot\boldsymbol{e}_{{\scriptscriptstyle 1}}^{\prime}=v_{{\scriptscriptstyle 1}}\\ \boldsymbol{e}^{{\scriptscriptstyle 2}}\left(\boldsymbol{v}\right)=\boldsymbol{e}^{{\scriptscriptstyle 2}}\cdot\boldsymbol{v}=\boldsymbol{v}\cdot\boldsymbol{e}_{{\scriptscriptstyle 2}}^{\prime}=v_{{\scriptscriptstyle 2}}\\ \boldsymbol{e}^{{\scriptscriptstyle 3}}\left(\boldsymbol{v}\right)=\boldsymbol{e}^{{\scriptscriptstyle 3}}\cdot\boldsymbol{v}=\boldsymbol{v}\cdot\boldsymbol{e}_{{\scriptscriptstyle 3}}^{\prime}=v_{{\scriptscriptstyle 3}} \end{cases}$

reciprocal space is a dual space of its original vector space

$\begin{aligned} V= & \mathrm{span}\left\{ \boldsymbol{e}_{{\scriptscriptstyle 1}},\boldsymbol{e}_{{\scriptscriptstyle 2}},\boldsymbol{e}_{{\scriptscriptstyle 3}}\right\} =\left\{ v_{{\scriptscriptstyle 1}}\boldsymbol{e}_{{\scriptscriptstyle 1}}+v_{{\scriptscriptstyle 2}}\boldsymbol{e}_{{\scriptscriptstyle 2}}+v_{{\scriptscriptstyle 3}}\boldsymbol{e}_{{\scriptscriptstyle 3}}\right\} \\ = & \left\{ \sum_{j=1}^{3}v_{{\scriptscriptstyle j}}\boldsymbol{e}_{{\scriptscriptstyle j}}\right\} =\left\{ v_{{\scriptscriptstyle j}}\boldsymbol{e}_{{\scriptscriptstyle j}}\middle|\begin{cases} v_{{\scriptscriptstyle j}}\in F\\ \boldsymbol{e}_{{\scriptscriptstyle j}}\in F^{3} \end{cases}\right\} =\left\{ \boldsymbol{v}\middle|\boldsymbol{v}\in V\right\} \\ V^{*}= & \mathrm{span}\left\{ \boldsymbol{e}^{{\scriptscriptstyle 1}},\boldsymbol{e}^{{\scriptscriptstyle 2}},\boldsymbol{e}^{{\scriptscriptstyle 3}}\right\} =\left\{ v^{*{\scriptscriptstyle 1}}\boldsymbol{e}^{{\scriptscriptstyle 1}}+v^{*{\scriptscriptstyle 2}}\boldsymbol{e}^{{\scriptscriptstyle 2}}+v^{*{\scriptscriptstyle 3}}\boldsymbol{e}^{{\scriptscriptstyle 3}}\right\} \\ = & \left\{ \sum_{i=1}^{3}v^{*{\scriptscriptstyle i}}\boldsymbol{e}^{{\scriptscriptstyle i}}\right\} =\left\{ v^{*{\scriptscriptstyle i}}\boldsymbol{e}^{{\scriptscriptstyle i}}\middle|\begin{cases} v^{*{\scriptscriptstyle i}}\in F\\ \boldsymbol{e}^{{\scriptscriptstyle i}}\in F^{3} \end{cases}\right\} =\left\{ \boldsymbol{v}^{*}\middle|\boldsymbol{v}^{*}\in V^{*}\right\} \end{aligned}$

$\begin{aligned} \boldsymbol{v}^{*}\left(\boldsymbol{v}\right)= & \left(v^{*{\scriptscriptstyle i}}\boldsymbol{e}^{{\scriptscriptstyle i}}\right)\left(\boldsymbol{v}\right),\boldsymbol{v}\in V\\ = & \left(v^{*{\scriptscriptstyle 1}}\boldsymbol{e}^{{\scriptscriptstyle 1}}+v^{*{\scriptscriptstyle 2}}\boldsymbol{e}^{{\scriptscriptstyle 2}}+v^{*{\scriptscriptstyle 3}}\boldsymbol{e}^{{\scriptscriptstyle 3}}\right)\left(\boldsymbol{v}\right)\\ = & v^{*{\scriptscriptstyle 1}}\boldsymbol{e}^{{\scriptscriptstyle 1}}\left(\boldsymbol{v}\right)+v^{*{\scriptscriptstyle 2}}\boldsymbol{e}^{{\scriptscriptstyle 2}}\left(\boldsymbol{v}\right)+v^{*{\scriptscriptstyle 3}}\boldsymbol{e}^{{\scriptscriptstyle 3}}\left(\boldsymbol{v}\right)\\ = & v^{*{\scriptscriptstyle 1}}v_{{\scriptscriptstyle 1}}+v^{*{\scriptscriptstyle 2}}v_{{\scriptscriptstyle 2}}+v^{*{\scriptscriptstyle 3}}v_{{\scriptscriptstyle 3}}\in F \end{aligned}$

element in dual space is a functional or mapping from its original vector space to the field

$\boldsymbol{v}^{*}:V\rightarrow F$

$V\overset{\boldsymbol{v}^{*}}{\rightarrow}F$

$V^{*}=\left\{ \boldsymbol{v}^{*}\middle|\boldsymbol{v}^{*}:V\rightarrow F\right\}$

$\begin{array}{cccccccccc} & & & V & =\{ & \boldsymbol{e}_{{\scriptscriptstyle 1}} & \boldsymbol{e}_{{\scriptscriptstyle 2}} & \boldsymbol{e}_{{\scriptscriptstyle 3}} & \boldsymbol{v} & \cdots\}\\ & \boldsymbol{e}^{{\scriptscriptstyle 1}} & : & \downarrow & & \downarrow & \downarrow & \downarrow & \downarrow\\ V^{*}=\{ & \boldsymbol{e}^{{\scriptscriptstyle 2}} & \} & F & \supseteq\{ & 1 & 0 & 0 & v_{{\scriptscriptstyle 1}} & \cdots\}\\ & \boldsymbol{e}^{{\scriptscriptstyle 3}} & & V & =\{ & \boldsymbol{e}_{{\scriptscriptstyle 1}} & \boldsymbol{e}_{{\scriptscriptstyle 2}} & \boldsymbol{e}_{{\scriptscriptstyle 3}} & \boldsymbol{v} & \cdots\}\\ & \boldsymbol{v}^{*} & : & \downarrow & & \downarrow & \downarrow & \downarrow & \downarrow\\ & \vdots & & F & \supseteq\{ & v^{*{\scriptscriptstyle 1}} & v^{*{\scriptscriptstyle 2}} & v^{*{\scriptscriptstyle 3}} & v^{*{\scriptscriptstyle i}}v_{{\scriptscriptstyle i}} & \cdots\} \end{array}$

$\begin{aligned} V^{*}=\left\{ \boldsymbol{v}^{*}\middle|\boldsymbol{v}^{*}\in V^{*}\right\} =&\left\{ \boldsymbol{v}^{*}\middle|\boldsymbol{v}^{*}:V\rightarrow F\right\} \\=&\left\{ \boldsymbol{v}^{*}\middle|V\overset{\boldsymbol{v}^{*}}{\rightarrow}F\right\} \\=&\left\{ \boldsymbol{\omega}\middle|\boldsymbol{\omega}:V\rightarrow F\right\} \\=&\left\{ \omega^{{\scriptscriptstyle i}}\boldsymbol{e}^{{\scriptscriptstyle i}}\middle|\begin{cases} \omega^{{\scriptscriptstyle i}}\in F\\ \boldsymbol{e}^{{\scriptscriptstyle i}}\in F^{3} \end{cases}\right\} \end{aligned}$

By defining vector addition and scalar multiplication on the dual space

$\begin{cases} +:V^{*}\times V^{*}\rightarrow V^{*}\Leftrightarrow\forall\boldsymbol{\omega}_{{\scriptscriptstyle 1}},\boldsymbol{\omega}_{{\scriptscriptstyle 2}}\in V^{*},\exists!\left(\boldsymbol{\omega}_{{\scriptscriptstyle 1}}+\boldsymbol{\omega}_{{\scriptscriptstyle 2}}\right)\in V^{*}\left[\left(\boldsymbol{\omega}_{{\scriptscriptstyle 1}}+\boldsymbol{\omega}_{{\scriptscriptstyle 2}}\right)\left(\boldsymbol{v}\right)=\boldsymbol{\omega}_{{\scriptscriptstyle 1}}\left(\boldsymbol{v}\right)+\boldsymbol{\omega}_{{\scriptscriptstyle 2}}\left(\boldsymbol{v}\right)\right]\\ \cdot:F\times V^{*}\rightarrow V^{*}\Leftrightarrow\forall k\in F,\forall\boldsymbol{\omega}\in V^{*},\exists!\left(k\boldsymbol{\omega}\right)\in V^{*}\left[\left(k\boldsymbol{\omega}\right)\left(\boldsymbol{v}\right)=k\cdot\boldsymbol{\omega}\left(\boldsymbol{v}\right)\right]\\ \forall\boldsymbol{\omega}\in V^{*},\exists!\boldsymbol{0}\in V^{*}\left[\left(\boldsymbol{\omega}+\boldsymbol{0}\right)\left(\boldsymbol{v}\right)=\boldsymbol{\omega}\left(\boldsymbol{v}\right)+\boldsymbol{0}\left(\boldsymbol{v}\right)=\boldsymbol{\omega}\left(\boldsymbol{v}\right)\right] \end{cases}$

the dual space also becomes a vector space.

43.4.1.2 double dual concept

double dual space = second dual space

$\begin{aligned} V^{**}= & \left(V^{*}\right)^{*}\\ = & \left\{ \boldsymbol{\omega}^{*}\middle|\boldsymbol{\omega}^{*}:V^{*}\rightarrow F\right\} \\ = & \left\{ \boldsymbol{\omega}^{*}\middle|\boldsymbol{\omega}^{*}\in V^{**}\right\} \end{aligned}$

$\begin{aligned} V^{**}=\left(V^{*}\right)^{*}= & \mathrm{span}\left\{ \boldsymbol{e}^{{\scriptscriptstyle \mu}}\right\} _{{\scriptscriptstyle \mu\in\left\{ 1,2,3\right\} }}^{*}\\ = & \mathrm{span}\left\{ \boldsymbol{e}^{{\scriptscriptstyle 1}},\boldsymbol{e}^{{\scriptscriptstyle 2}},\boldsymbol{e}^{{\scriptscriptstyle 3}}\right\} ^{*}\\ = & \mathrm{span}\left\{ \boldsymbol{e}^{{\scriptscriptstyle 1}*},\boldsymbol{e}^{{\scriptscriptstyle 2}*},\boldsymbol{e}^{{\scriptscriptstyle 3}*}\right\} \\ = & \mathrm{span}\left\{ \boldsymbol{e}^{{\scriptscriptstyle \nu}*}\right\} _{{\scriptscriptstyle \nu\in\left\{ 1,2,3\right\} }} \end{aligned}$

$\begin{aligned} \boldsymbol{\omega}^{*}\left(\boldsymbol{\omega}\right)= & \left(\omega^{*{\scriptscriptstyle \nu}}\boldsymbol{e}^{{\scriptscriptstyle \nu}*}\right)\left(\boldsymbol{\omega}\right),\boldsymbol{\omega}\in V^{*}\\ = & \left(\omega^{*{\scriptscriptstyle 1}}\boldsymbol{e}^{{\scriptscriptstyle 1}*}+\omega^{*{\scriptscriptstyle 2}}\boldsymbol{e}^{{\scriptscriptstyle 2}*}+\omega^{*{\scriptscriptstyle 3}}\boldsymbol{e}^{{\scriptscriptstyle 3}*}\right)\left(\boldsymbol{\omega}\right)\\ = & \omega^{*{\scriptscriptstyle 1}}\boldsymbol{e}^{{\scriptscriptstyle 1}*}\left(\boldsymbol{\omega}\right)+\omega^{*{\scriptscriptstyle 2}}\boldsymbol{e}^{{\scriptscriptstyle 2}*}\left(\boldsymbol{\omega}\right)+\omega^{*{\scriptscriptstyle 3}}\boldsymbol{e}^{{\scriptscriptstyle 3}*}\left(\boldsymbol{\omega}\right)\\ = & \omega^{*{\scriptscriptstyle 1}}\omega_{{\scriptscriptstyle 1}}+\omega^{*{\scriptscriptstyle 2}}\omega_{{\scriptscriptstyle 2}}+\omega^{*{\scriptscriptstyle 3}}\omega_{{\scriptscriptstyle 3}}\in F \end{aligned}$

$V^{**}=\left\{ \boldsymbol{\omega}^{*}\middle|\boldsymbol{\omega}^{*}:V^{*}\rightarrow F\right\}$

$\begin{cases} \boldsymbol{e}^{{\scriptscriptstyle 1}*}\left(\boldsymbol{\omega}\right)=\boldsymbol{e}^{{\scriptscriptstyle 1}*}\cdot\boldsymbol{\omega}=\boldsymbol{\omega}\cdot\boldsymbol{e}^{{\scriptscriptstyle 1}*}=\boldsymbol{\omega}\left(\boldsymbol{e}^{{\scriptscriptstyle 1}*}\right)\\ \boldsymbol{e}^{{\scriptscriptstyle 2}*}\left(\boldsymbol{\omega}\right)=\boldsymbol{e}^{{\scriptscriptstyle 2}*}\cdot\boldsymbol{\omega}=\boldsymbol{\omega}\cdot\boldsymbol{e}^{{\scriptscriptstyle 2}*}=\boldsymbol{\omega}\left(\boldsymbol{e}^{{\scriptscriptstyle 2}*}\right)\\ \boldsymbol{e}^{{\scriptscriptstyle 3}*}\left(\boldsymbol{\omega}\right)=\boldsymbol{e}^{{\scriptscriptstyle 3}*}\cdot\boldsymbol{\omega}=\boldsymbol{\omega}\cdot\boldsymbol{e}^{{\scriptscriptstyle 3}*}=\boldsymbol{\omega}\left(\boldsymbol{e}^{{\scriptscriptstyle 3}*}\right) \end{cases}$

$\boldsymbol{\omega}^{*}\left(\boldsymbol{\omega}\right)=\boldsymbol{\omega}^{*}\cdot\boldsymbol{\omega}=\boldsymbol{\omega}\cdot\boldsymbol{\omega}^{*}=\boldsymbol{\omega}\left(\boldsymbol{\omega}^{*}\right)$ i.e. $\boldsymbol{f}$ acts on $\boldsymbol{x}$ equivalent to $\boldsymbol{x}$ acts on $\boldsymbol{f}$

$\boldsymbol{x}\left(\boldsymbol{f}\right)=\boldsymbol{x}\cdot\boldsymbol{f}=\boldsymbol{f}\cdot\boldsymbol{x}=\boldsymbol{f}\left(\boldsymbol{x}\right)$

$\boldsymbol{e}^{{\scriptscriptstyle \mu}}\left(\boldsymbol{e}_{{\scriptscriptstyle \nu}}\right)=\boldsymbol{e}^{{\scriptscriptstyle \mu}}\cdot\boldsymbol{e}_{{\scriptscriptstyle \nu}}=\delta_{{\scriptscriptstyle \nu}}^{{\scriptscriptstyle \mu}}=\delta^{{\scriptscriptstyle \mu}}{}_{{\scriptscriptstyle \nu}}=\begin{cases} 1 & \mu=\nu\\ 0 & \mu\ne\nu \end{cases}$

$\begin{aligned} \boldsymbol{e}^{{\scriptscriptstyle \nu}*}\left(\boldsymbol{e}^{{\scriptscriptstyle \mu}}\right)\overset{\text{def.}}{=}\boldsymbol{e}_{{\scriptscriptstyle \nu}}\cdot\boldsymbol{e}^{{\scriptscriptstyle \mu}}= & \boldsymbol{e}^{{\scriptscriptstyle \mu}}\cdot\boldsymbol{e}_{{\scriptscriptstyle \nu}}=\boldsymbol{e}^{{\scriptscriptstyle \mu}}\left(\boldsymbol{e}_{{\scriptscriptstyle \nu}}\right)\\ \Downarrow\\ V^{**}=\mathrm{span}\left\{ \boldsymbol{e}^{{\scriptscriptstyle \nu}*}\right\} _{{\scriptscriptstyle \nu\in\left\{ 1,2,3\right\} }}\cong & \mathrm{span}\left\{ \boldsymbol{e}_{{\scriptscriptstyle \nu}}\right\} _{{\scriptscriptstyle \nu\in\left\{ 1,2,3\right\} }}=V\\ V^{**}\cong & V\\ \Downarrow & \begin{cases} V^{**}\cong V & V,V^{**}\text{ are isomorphism}\\ & \text{independent of choice of bases} \end{cases}\\ V,V^{**} & \text{ are naturally isomorphism} \end{aligned}$

$V^{**}=\left\{ \boldsymbol{\omega}^{*}\middle|\boldsymbol{\omega}^{*}:V^{*}\rightarrow F\right\} \cong V=\left\{ v\middle|v:V^{*}\rightarrow F\right\}$

$\begin{array}{cccccccccc} & & & V^{*} & =\{ & \boldsymbol{e}^{{\scriptscriptstyle 1}} & \boldsymbol{e}^{{\scriptscriptstyle 2}} & \boldsymbol{e}^{{\scriptscriptstyle 3}} & \boldsymbol{\omega} & \cdots\}\\ & \boldsymbol{e}^{{\scriptscriptstyle 1}*} & : & \downarrow & & \downarrow & \downarrow & \downarrow & \downarrow\\ V^{**}=\{ & \boldsymbol{e}^{{\scriptscriptstyle 2}*} & \} & F & \supseteq\{ & 1 & 0 & 0 & \omega^{{\scriptscriptstyle 1}} & \cdots\}\\ & \boldsymbol{e}^{{\scriptscriptstyle 3}*} & & V^{*} & =\{ & \boldsymbol{e}^{{\scriptscriptstyle 1}} & \boldsymbol{e}^{{\scriptscriptstyle 2}} & \boldsymbol{e}^{{\scriptscriptstyle 3}} & \boldsymbol{\omega} & \cdots\}\\ & \boldsymbol{\omega}^{*} & : & \downarrow & & \downarrow & \downarrow & \downarrow & \downarrow\\ & \vdots & & F & \supseteq\{ & \omega^{{\scriptscriptstyle 1}*} & \omega^{{\scriptscriptstyle 2}*} & \omega^{{\scriptscriptstyle 3}*} & \omega^{{\scriptscriptstyle \mu}*}\omega^{{\scriptscriptstyle \mu}} & \cdots\}\\ & & & V^{*} & =\{ & \boldsymbol{e}^{{\scriptscriptstyle 1}} & \boldsymbol{e}^{{\scriptscriptstyle 2}} & \boldsymbol{e}^{{\scriptscriptstyle 3}} & \boldsymbol{v}^{*} & \cdots\}\\ & \boldsymbol{e}_{{\scriptscriptstyle 1}} & : & \downarrow & & \downarrow & \downarrow & \downarrow & \downarrow\\ \cong V=\{ & \boldsymbol{e}_{{\scriptscriptstyle 2}} & \} & F & \supseteq\{ & 1 & 0 & 0 & v^{*{\scriptscriptstyle 1}} & \cdots\}\\ & \boldsymbol{e}_{{\scriptscriptstyle 3}} & & V^{*} & =\{ & \boldsymbol{e}^{{\scriptscriptstyle 1}} & \boldsymbol{e}^{{\scriptscriptstyle 2}} & \boldsymbol{e}^{{\scriptscriptstyle 3}} & \boldsymbol{v}^{*} & \cdots\}\\ & \boldsymbol{v} & : & \downarrow & & \downarrow & \downarrow & \downarrow & \downarrow\\ & \vdots & & F & \supseteq\{ & v_{{\scriptscriptstyle 1}} & v_{{\scriptscriptstyle 2}} & v_{{\scriptscriptstyle 3}} & v_{{\scriptscriptstyle \mu}}v^{*{\scriptscriptstyle \mu}} & \cdots\} \end{array}$

$V\cong V^{**}$

$V\cong V^{**}=\left\{ \boldsymbol{\omega}^{*}\middle|\boldsymbol{\omega}^{*}:V^{*}\rightarrow F\right\}$

$V=\left\{ v\middle|v:V^{*}\rightarrow F\right\}$

i.e. vector space is a set of functionals or mappings from its dual space to the field, answering What is a vector?^[43.1], and satifying Fig: 43.1.