Physlib.Relativity.Tensors.RealTensor.Vector.Basic

For a given spatial dimension $d$, the space of Lorentz vectors $\text{Vector}_d$, which are defined as functions from the index set $\text{Fin } 1 \oplus \text{Fin } d$ to the real numbers $\mathbb{R}$, forms an additive commutative monoid. This structure provides an associative and commutative addition operation and a zero element.

For a given spatial dimension $d$, the space of Lorentz vectors $\text{Vector}_d$, defined as the set of functions mapping the index set $\text{Fin } 1 \oplus \text{Fin } d$ to the real numbers $\mathbb{R}$, forms a vector space (module) over the field of real numbers $\mathbb{R}$.

For a given spatial dimension $d$, the space of Lorentz vectors $\text{Vector}_d$, defined as the set of functions from the index set $\text{Fin } 1 \oplus \text{Fin } d$ to the real numbers $\mathbb{R}$, forms an additive commutative group. This structure defines the addition of vectors, the existence of a zero vector, and the existence of additive inverses (negation).

For any spatial dimension $d$, the space of Lorentz vectors $\text{Vector}_d$, defined as the set of functions from the index set $\text{Fin } 1 \oplus \text{Fin } d$ to the real numbers $\mathbb{R}$, is a finite-dimensional vector space over $\mathbb{R}$.

For a given spatial dimension $d$, the map `equivEuclid` defines a linear isomorphism (linear equivalence) between the space of Lorentz vectors $\text{Vector}_d$ and the Euclidean space $\mathbb{R}^{1+d}$, specifically represented as $\text{EuclideanSpace } \mathbb{R} (\text{Fin } 1 \oplus \text{Fin } d)$. This equivalence identifies the components of a Lorentz vector with the coordinates of a vector in $(1+d)$-dimensional Euclidean space.

For any spatial dimension $d \in \mathbb{N}$, Lorentz vector $v \in \text{Vector}_d$, and index $i \in \text{Fin } 1 \oplus \text{Fin } d$, the value of the linear isomorphism $\text{equivEuclid } d$ applied to $v$ and evaluated at index $i$ is equal to the $i$-th component of $v$, i.e., $(\text{equivEuclid } d (v))_i = v_i$.

For a given spatial dimension $d$, the norm $\|\cdot\|$ on the space of Lorentz vectors $\text{Vector}_d$ is defined such that for any $v \in \text{Vector}_d$, its norm $\|v\|$ is equal to the standard Euclidean norm of its image under the linear isomorphism $\text{equivEuclid}_d : \text{Vector}_d \cong \mathbb{R}^{1+d}$. This defines a positive-definite norm based on the components of the vector in the underlying Euclidean space.

For any spatial dimension $d \in \mathbb{N}$ and any Lorentz vector $v \in \text{Vector}_d$, the norm $\|v\|$ is equal to the Euclidean norm of its image under the linear isomorphism $\text{equivEuclid}_d$, which maps the space of Lorentz vectors to the Euclidean space $\mathbb{R}^{1+d}$ (represented as $\text{EuclideanSpace } \mathbb{R} (\text{Fin } 1 \oplus \text{Fin } d)$).

For any Lorentz vector $v$ in $1+d$ dimensions and any index $i$ in the set $\text{Fin } 1 \oplus \text{Fin } d$, the absolute value of the $i$-th component of the vector is less than or equal to its Euclidean norm, i.e., $|v_i| \leq \|v\|$. Here, the norm $\|v\|$ is defined as the standard Euclidean norm of the vector's components in $\mathbb{R}^{1+d}$.

For a given spatial dimension $d$, the space of Lorentz vectors $\text{Vector}_d$ is a normed additive commutative group. This structure integrates the additive commutative group properties of $\text{Vector}_d$ with its Euclidean norm $\|\cdot\|$, such that the distance between two vectors $x, y \in \text{Vector}_d$ is defined by the norm of their difference, $dist(x, y) = \|x - y\|$.

For a given spatial dimension $d$, the space of Lorentz vectors $\text{Vector}_d$ is a normed vector space over the field of real numbers $\mathbb{R}$. This structure signifies that the Euclidean norm $\|\cdot\|$ on $\text{Vector}_d$ is compatible with scalar multiplication, such that for any scalar $\alpha \in \mathbb{R}$ and any vector $v \in \text{Vector}_d$, the property $\|\alpha v\| = |\alpha| \|v\|$ holds.

For a given spatial dimension $d$, the space of Lorentz vectors $\text{Vector}_d$ is equipped with an inner product structure over the real numbers $\mathbb{R}$. The inner product of two Lorentz vectors $v, w \in \text{Vector}_d$, denoted by $\langle v, w \rangle_{\mathbb{R}}$, is defined as the standard Euclidean inner product of their representations in the $(1+d)$-dimensional Euclidean space $\mathbb{R}^{1+d}$ via the linear equivalence `equivEuclid`. This inner product is distinct from the Minkowski inner product and is used to define the topological and normed space structure of the vector space.

For any spatial dimension $d \in \mathbb{N}$ and any two Lorentz vectors $v, w \in \text{Vector}_d$, the real inner product $\langle v, w \rangle_{\mathbb{R}}$ is equal to the standard Euclidean inner product of their images under the linear equivalence $\text{equivEuclid} \colon \text{Vector}_d \cong \mathbb{R}^{1+d}$. That is, $\langle v, w \rangle_{\mathbb{R}} = \langle \text{equivEuclid}(v), \text{equivEuclid}(w) \rangle_{\mathbb{R}}$.

For a given spatial dimension $d \in \mathbb{N}$, the space of Lorentz vectors $\text{Vector}_d$ is equipped with the structure of an inner product space over the real numbers $\mathbb{R}$. This inner product $\langle \cdot, \cdot \rangle_{\mathbb{R}}$ is the standard Euclidean inner product (distinct from the Minkowski metric) and is defined via the linear equivalence between $\text{Vector}_d$ and the Euclidean space $\mathbb{R}^{1+d}$. It satisfies the axioms of a real inner product space, including symmetry, linearity in the first argument, and the identity $\|v\|^2 = \langle v, v \rangle_{\mathbb{R}}$ for any vector $v \in \text{Vector}_d$.

The space of Lorentz vectors $\text{Vector}_d$ is a charted space (manifold) modeled on the vector space $\text{Vector}_d$ itself. This structure equips the space with a manifold atlas where the local charts are identity maps from $\text{Vector}_d$ to $\text{Vector}_d$.

A Lorentz vector $v$ in $d$ spatial dimensions can be treated as a function that maps an index $i$ from the set $\{0, 1, \dots, d\}$ (represented formally by the sum type $\text{Fin } 1 \oplus \text{Fin } d$) to its corresponding real-valued component $v_i \in \mathbb{R}$. This coercion allows the notation $v(i)$ to be used to access the $i$-th component of the vector.

For any spatial dimension $d$, given a scalar $c \in \mathbb{R}$, a Lorentz vector $v \in \text{Vector}_d$, and an index $i \in \{0, 1, \dots, d\}$ (represented by the sum type $\text{Fin } 1 \oplus \text{Fin } d$), the $i$-th component of the scalar multiple $c \cdot v$ is equal to the product of $c$ and the $i$-th component of $v$. That is, $(c \cdot v)_i = c \cdot v_i$.

For any spatial dimension $d$, given two Lorentz vectors $v$ and $w$ in $\text{Vector}_d$ and an index $i \in \{0, 1, \dots, d\}$, the $i$-th component of the sum of the vectors is equal to the sum of their individual $i$-th components. That is, $(v + w)_i = v_i + w_i$.

For any spatial dimension $d \in \mathbb{N}$, given two Lorentz vectors $v, w \in \text{Vector}_d$ and an index $i \in \{0, 1, \dots, d\}$, the $i$-th component of the difference of the vectors is equal to the difference of their individual $i$-th components. That is, $$(v - w)_i = v_i - w_i$$

For any spatial dimension $d \in \mathbb{N}$, given a finite index set $\iota$, a family of Lorentz vectors $f : \iota \to \text{Vector}_d$, and an index $i \in \{0, 1, \dots, d\}$, the $i$-th component of the sum of these vectors is equal to the sum of the $i$-th components of each individual vector. That is, $$\left( \sum_{j \in \iota} f(j) \right)_i = \sum_{j \in \iota} f(j)_i$$

For any spatial dimension $d \in \mathbb{N}$, given a Lorentz vector $v \in \text{Vector}_d$ and an index $i \in \text{Fin } 1 \oplus \text{Fin } d$, the $i$-th component of the negated vector $-v$ is equal to the negation of the $i$-th component of $v$. That is, $$(-v)_i = -v_i$$

For any spatial dimension $d \in \mathbb{N}$ and any index $i \in \text{Fin } 1 \oplus \text{Fin } d$, the $i$-th component of the zero Lorentz vector $0 \in \text{Vector}_d$ is $0$.

For a spatial dimension $d \in \mathbb{N}$ and an index $i \in \text{Fin } 1 \oplus \text{Fin } d$, this is the continuous linear map from the space of Lorentz vectors $\text{Vector}_d$ to the real numbers $\mathbb{R}$ that maps each vector $v$ to its $i$-th component $v_i$.

For a spatial dimension $d \in \mathbb{N}$, an index $i \in \text{Fin } 1 \oplus \text{Fin } d$, and a Lorentz vector $v \in \text{Vector}_d$, the application of the continuous linear map $\text{coordCLM}_i$ to the vector $v$ is equal to its $i$-th component $v_i$.

For a spatial dimension $d \in \mathbb{N}$ and an index $i \in \text{Fin } 1 \oplus \text{Fin } d$, the map $v \mapsto v_i$ from the space of Lorentz vectors $\text{Vector}_d$ to the real numbers $\mathbb{R}$ is continuous.

For any spatial dimension $d \in \mathbb{N}$ and any index $i \in \text{Fin } 1 \oplus \text{Fin } d$, the coordinate map from the space of Lorentz vectors $\text{Vector}_d$ to the real numbers $\mathbb{R}$, defined by $v \mapsto v_i$, is $n$-times continuously differentiable ($C^n$) over $\mathbb{R}$ for any $n$.

For a spatial dimension $d \in \mathbb{N}$ and an index $i \in \text{Fin } 1 \oplus \text{Fin } d$, the coordinate function $v \mapsto v_i$, which maps a Lorentz vector $v \in \text{Vector}_d$ to its $i$-th component in $\mathbb{R}$, is differentiable over $\mathbb{R}$.

For a given spatial dimension $d \in \mathbb{N}$, an index $i \in \text{Fin } 1 \oplus \text{Fin } d$, and a Lorentz vector $v \in \text{Vector}_d$, the map $v \mapsto v_i$ that sends a vector to its $i$-th component is differentiable at $v$ over the field of real numbers $\mathbb{R}$.

For a given spatial dimension $d$, the map `euclidCLE` defines a continuous linear equivalence (a topological vector space isomorphism) between the space of Lorentz vectors $\text{Vector}_d$ and the $(1+d)$-dimensional Euclidean space, represented as $\text{EuclideanSpace } \mathbb{R} (\text{Fin } 1 \oplus \text{Fin } d)$. This map is the continuous version of the linear equivalence `equivEuclid`.

For a given spatial dimension $d$, this definition provides the continuous linear equivalence between the space of Lorentz vectors $\text{Vector}_d$ and the space of functions (the product type) from the index set $\text{Fin } 1 \oplus \text{Fin } d$ to the real numbers $\mathbb{R}$, denoted as $\prod_{i \in \text{Fin } 1 \oplus \text{Fin } d} \mathbb{R}$. This map is a bijective linear transformation that is continuous in both directions, identifying Lorentz vectors with their component representations.

For any spatial dimension $d$, let $v \in \text{Vector}_d$ be a Lorentz vector and $i \in \text{Fin } 1 \oplus \text{Fin } d$ be an index. The application of the continuous linear equivalence $\text{equivPi}$ to $v$, evaluated at the index $i$, is equal to the $i$-th component $v_i$.

Let $d$ be a natural number and $\alpha$ be a topological space. For a function $f : \alpha \to \text{Vector}_d$ mapping into the space of Lorentz vectors, if for every index $i \in \text{Fin } 1 \oplus \text{Fin } d$ the component function $x \mapsto f(x)_i$ is continuous, then the function $f$ is continuous.

For a spatial dimension $d \in \mathbb{N}$, let $\text{Vector}_d$ be the space of Lorentz vectors. Let $\alpha$ be a normed space over $\mathbb{R}$. A function $f : \alpha \to \text{Vector}_d$ is differentiable if and only if for every index $i \in \text{Fin } 1 \oplus \text{Fin } d$, the component function $f_i : \alpha \to \mathbb{R}$ (defined by $x \mapsto (f(x))_i$) is differentiable.

Let $\alpha$ be a real normed vector space and $d$ be a natural number representing the spatial dimensions. For any $n \in \mathbb{N} \cup \{\infty\}$, a function $f: \alpha \to \text{Vector}_d$ is $n$-times continuously differentiable ($C^n$) if and only if for every index $i \in \text{Fin } 1 \oplus \text{Fin } d$, the scalar-valued component function $x \mapsto f(x)_i$ is $n$-times continuously differentiable.

Let $d \in \mathbb{N}$ be the spatial dimension and $\alpha$ be a real normed space. For any differentiable function $f: \alpha \to \text{Vector}_d$, any point $x \in \alpha$, direction $dt \in \alpha$, and index $\nu \in \text{Fin } 1 \oplus \text{Fin } d$, the $\nu$-th component of the Fréchet derivative of $f$ at $x$ in the direction $dt$ is equal to the Fréchet derivative of the component function $f_\nu: \alpha \to \mathbb{R}$ (where $f_\nu(y) = f(y)_\nu$) at $x$ in the direction $dt$. Mathematically, this is expressed as: \[ (Df(x) \cdot dt)_\nu = D(f_\nu)(x) \cdot dt \]

For a given number of spatial dimensions $d \in \mathbb{N}$ and an index $\mu \in \text{Fin } 1 \oplus \text{Fin } d$, the Fréchet derivative of the function that maps a Lorentz vector $v \in \text{Vector}_d$ to its $\mu$-th component $v_\mu$ is equal to the continuous linear map $\text{coordCLM } \mu$, which evaluates the $\mu$-th coordinate of its argument.

For a spacetime with $d$ spatial dimensions, the type of component indices for a Lorentz vector (a tensor with a single contravariant index) is equivalent to the disjoint union $\text{Fin}(1) \oplus \text{Fin}(d)$. This equivalence maps the single index of the vector to either the temporal component (index $0$ in $\text{Fin}(1)$) or one of the $d$ spatial components (indices in $\text{Fin}(d)$).

For a given number of spatial dimensions $d \in \mathbb{N}$, this definition establishes that the space of Lorentz vectors $\text{Vector}_d$ is tensorial with respect to the species of real Lorentz tensors with a single contravariant (upper) index. This is achieved by providing a linear isomorphism between $\text{Vector}_d$ and the formal tensor space associated with the color sequence $[ \text{up} ]$. Specifically, the isomorphism identifies a Lorentz vector with its components in the canonical tensor basis, where the index set is equivalent to the disjoint union $\text{Fin}(1) \oplus \text{Fin}(d)$, representing the temporal and spatial components respectively.

Let $d$ be the number of spatial dimensions. For any real Lorentz tensor $p$ with a single contravariant (upper) index, the image of $p$ under the inverse isomorphism $\text{toTensor}^{-1}$ (which maps tensors back to the space of Lorentz vectors $\text{Vector}_d$) is the function that assigns to each index $i \in \text{Fin } 1 \oplus \text{Fin } d$ the coordinate of $p$ with respect to the canonical tensor basis, where the tensor's multi-indices are identified with $\text{Fin } 1 \oplus \text{Fin } d$ via the index equivalence map.

Let $d$ be the number of spatial dimensions. For any pure tensor $p$ of rank 1 with a single contravariant (upper) index, let $p_0$ be its constituent vector in the representation space. For any index $i \in \text{Fin } 1 \oplus \text{Fin } d$, the $i$-th component of the Lorentz vector obtained from $p$ via the inverse isomorphism $\text{toTensor}^{-1}$ is given by: $$(\text{toTensor}^{-1}(p.\text{toTensor}))_i = \text{repr}(p_0)_{j}$$ where $\text{repr}(p_0)$ denotes the coordinate representation of the vector $p_0$ with respect to the standard contravariant basis, and $j$ is the index in the representation space corresponding to $i$ via the index equivalence map.

For a given spatial dimension $d \in \mathbb{N}$, this provides the standard coordinate basis for the space of Lorentz vectors $\text{Vector}_d$ over the field of real numbers $\mathbb{R}$. The basis is indexed by the set $\text{Fin } 1 \oplus \text{Fin } d$, which represents the combination of one temporal and $d$ spatial dimensions. A basis vector indexed by $\mu$ corresponds to the function that yields $1$ at index $\mu$ and $0$ for all other indices $\nu$.

For a given spatial dimension $d \in \mathbb{N}$, let $\text{basis } \mu$ denote the $\mu$-th vector of the standard coordinate basis for the space of Lorentz vectors $\text{Vector}_d$, where the indices $\mu, \nu$ belong to the set $\text{Fin } 1 \oplus \text{Fin } d$. The $\nu$-th component of the $\mu$-th basis vector is given by: $$(\text{basis } \mu)_\nu = \begin{cases} 1 & \text{if } \mu = \nu \\ 0 & \text{if } \mu \neq \nu \end{cases}$$

For a spacetime with $d$ spatial dimensions, let $\text{Vector}_d$ be the space of Lorentz vectors and let $\text{Tensor}_{\text{up}}$ be the space of real Lorentz tensors with a single contravariant (upper) index. Let $\text{toTensor} : \text{Vector}_d \cong \text{Tensor}_{\text{up}}$ be the linear isomorphism that identifies a Lorentz vector with its tensorial representation. Let $e_\mu$ be the $\mu$-th basis vector of the standard coordinate basis for $\text{Vector}_d$, where $\mu \in \text{Fin } 1 \oplus \text{Fin } d$, and let $E_b$ be the canonical basis for the tensor space indexed by component indices $b$. Then, the inverse of the isomorphism $\text{toTensor}$ maps the tensor basis element associated with $\mu$ to the corresponding Lorentz vector basis vector: $$\text{toTensor}^{-1}(E_{\phi(\mu)}) = e_\mu$$ where $\phi$ is the equivalence map (`indexEquiv.symm`) from the physical index set $\text{Fin } 1 \oplus \text{Fin } d$ to the tensor component index set.

For a spacetime with $d$ spatial dimensions, let $\text{Vector}_d$ be the space of Lorentz vectors and let $\text{Tensor}_{\text{up}}$ be the space of real Lorentz tensors with a single contravariant (upper) index. Let $\text{toTensor} : \text{Vector}_d \cong \text{Tensor}_{\text{up}}$ be the linear isomorphism that identifies a Lorentz vector with its tensorial representation. Let $e_\mu$ be the $\mu$-th basis vector of the standard coordinate basis for $\text{Vector}_d$, where $\mu \in \text{Fin } 1 \oplus \text{Fin } d$, and let $E_b$ be the canonical basis for the tensor space indexed by component indices $b$. Then, the isomorphism $\text{toTensor}$ maps the $\mu$-th basis vector of the Lorentz vector space to the corresponding basis vector in the tensor space: $$\text{toTensor}(e_\mu) = E_{\phi(\mu)}$$ where $\phi$ is the equivalence map (`indexEquiv.symm`) from the physical index set $\text{Fin } 1 \oplus \text{Fin } d$ to the tensor component index set.

For a spacetime with $d$ spatial dimensions, let $\text{Vector}_d$ be the space of Lorentz vectors and $\{e_\mu\}_{\mu \in \text{Fin } 1 \oplus \text{Fin } d}$ be its standard coordinate basis. Let $\text{Tensor}_{\text{up}}$ be the space of real Lorentz tensors with a single contravariant (upper) index, and let $\{E_b\}$ be its canonical basis indexed by component indices $b$. Given the linear isomorphism $\text{toTensor} : \text{Vector}_d \cong \text{Tensor}_{\text{up}}$ and the index equivalence $\phi : \text{ComponentIdx}(\text{up}) \cong \text{Fin } 1 \oplus \text{Fin } d$, the basis for Lorentz vectors is equal to the tensor basis mapped through the inverse isomorphism $\text{toTensor}^{-1}$ and reindexed by $\phi$: $$e = \text{reindex}_\phi (\text{toTensor}^{-1}(E))$$ Specifically, for each $\mu \in \text{Fin } 1 \oplus \text{Fin } d$, the basis vector $e_\mu$ is the image of the tensor basis element $E_{\phi^{-1}(\mu)}$ under the map $\text{toTensor}^{-1}$.

For a spacetime with $d$ spatial dimensions, let $\text{Vector}_d$ be the space of Lorentz vectors and let $T_{\text{up}}$ be the space of real Lorentz tensors with a single contravariant (upper) index. Let $\text{toTensor} : \text{Vector}_d \cong T_{\text{up}}$ be the canonical linear isomorphism between these spaces. Let $E$ be the canonical basis of $T_{\text{up}}$ indexed by $b \in \text{ComponentIdx}(\text{up})$, and let $e$ be the standard coordinate basis of $\text{Vector}_d$ indexed by $\mu \in \text{Fin } 1 \oplus \text{Fin } d$. Given the index equivalence $\phi : \text{ComponentIdx}(\text{up}) \cong \text{Fin } 1 \oplus \text{Fin } d$, the image of the tensor basis under the inverse isomorphism $\text{toTensor}^{-1}$ is equal to the Lorentz vector basis reindexed by the inverse equivalence $\phi^{-1}$: $$\text{toTensor}^{-1}(E) = \text{reindex}_{\phi^{-1}}(e)$$ Specifically, for each tensor component index $b$, the image of the $b$-th tensor basis vector under $\text{toTensor}^{-1}$ is the $\phi(b)$-th Lorentz vector basis vector.

For a spacetime with $d$ spatial dimensions, let $p \in \text{Vector}_d$ be a Lorentz vector and let $\text{toTensor}(p)$ be its representation as a rank-1 tensor with a single contravariant (upper) index. For any tensor component index $\mu$, the coordinate of $\text{toTensor}(p)$ with respect to the canonical tensor basis at index $\mu$ is equal to the value of the vector $p$ evaluated at the corresponding spacetime index $\text{indexEquiv}(\mu) \in \text{Fin } 1 \oplus \text{Fin } d$.

For any spatial dimension $d \in \mathbb{N}$, let $p$ be a Lorentz vector in $\text{Vector}_d$ and $\mu \in \text{Fin } 1 \oplus \text{Fin } d$ be a spacetime index. The $\mu$-th coordinate of $p$ with respect to the standard basis is equal to the component value $p(\mu)$.

For a spatial dimension $d \in \mathbb{N}$, let $f: \text{Vector}_d \to \text{Vector}_d$ be an $\mathbb{R}$-linear map and $p \in \text{Vector}_d$ be a Lorentz vector. The application of the map $f$ to the vector $p$ is equal to the matrix-vector product of the matrix representation of $f$ (with respect to the standard basis) and the vector $p$. That is, $f(p) = M_f \cdot p$, where $M_f$ is the matrix of $f$ relative to the standard basis.

For a spatial dimension $d \in \mathbb{N}$, let $f: \text{Fin } 1 \oplus \text{Fin } d \to \mathbb{R}$ be a collection of real coefficients. The linear combination of the standard basis vectors for the space of Lorentz vectors, $\sum_{\mu} f(\mu) \cdot \text{basis}_{\mu}$, is equal to zero if and only if $f(\mu) = 0$ for all $\mu \in \text{Fin } 1 \oplus \text{Fin } d$.

For a spatial dimension $d \in \mathbb{N}$, let $f_0 \in \mathbb{R}$ and $f: \text{Fin } d \to \mathbb{R}$ be coefficients. The linear combination of the temporal basis vector and the spatial basis vectors in the space of Lorentz vectors $\text{Vector}_d$, given by $f_0 \cdot \text{basis}(\text{inl } 0) + \sum_{i} f_i \cdot \text{basis}(\text{inr } i)$, is equal to zero if and only if $f_0 = 0$ and $f_i = 0$ for all $i \in \text{Fin } d$.

For a spacetime with $d$ spatial dimensions, let $\Lambda$ be a Lorentz transformation and $p$ be a Lorentz vector in $\text{Vector}_d$. For any index $i \in \text{Fin } 1 \oplus \text{Fin } d$, the $i$-th component of the vector resulting from the action of $\Lambda$ on $p$ is given by the sum over all indices $j$: $$(\Lambda \cdot p)_i = \sum_{j} \Lambda_{ij} p_j$$ where $\Lambda_{ij}$ represents the $(i, j)$-th entry of the matrix representation of the Lorentz transformation $\Lambda$, and $p_j$ is the $j$-th component of the vector $p$.

For any spatial dimension $d$, let $\Lambda$ be a Lorentz transformation in the Lorentz group and $p$ be a Lorentz vector in $\text{Vector}_d$. The action of $\Lambda$ on $p$, denoted by $\Lambda \cdot p$, is equal to the matrix-vector product $\Lambda \mathbf{v}_p$, where $\Lambda$ is represented by its underlying matrix and $\mathbf{v}_p$ is the vector of components of $p$.

For any spatial dimension $d \in \mathbb{N}$, let $\Lambda$ be a Lorentz transformation in the Lorentz group and $p, q$ be two Lorentz vectors in $\text{Vector}_d$. The action of $\Lambda$ on the sum of the two vectors satisfies the distributive property, such that $\Lambda \cdot (p + q) = \Lambda \cdot p + \Lambda \cdot q$.

For any spatial dimension $d \in \mathbb{N}$, let $\Lambda$ be a Lorentz transformation in the Lorentz group and $p, q$ be two Lorentz vectors in $\text{Vector}_d$. The action of $\Lambda$ on the difference of the two vectors satisfies the distributive property, such that $\Lambda \cdot (p - q) = \Lambda \cdot p - \Lambda \cdot q$.

For a spacetime with $d$ spatial dimensions, let $\Lambda$ be a Lorentz transformation in the Lorentz group and $0 \in \text{Vector}_d$ be the zero Lorentz vector. The action of $\Lambda$ on the zero vector is the zero vector: $$\Lambda \cdot 0 = 0$$

For a spacetime with $d$ spatial dimensions, let $\Lambda$ be a Lorentz transformation in the Lorentz group and $p$ be a Lorentz vector. The action of $\Lambda$ on the additive inverse (negation) of $p$ is equal to the negation of the result of $\Lambda$ acting on $p$: $$\Lambda \cdot (-p) = -(\Lambda \cdot p)$$

For a spacetime with $d$ spatial dimensions, let $\Lambda$ be a Lorentz transformation and $p$ be a Lorentz vector. The action of the negated Lorentz transformation $-\Lambda$ on the vector $p$ is equal to the negation of the result of the action of $\Lambda$ on $p$: $$(-\Lambda) \cdot p = -(\Lambda \cdot p)$$

For a given spatial dimension $d$ and a Lorentz transformation $\Lambda$ in the Lorentz group, this definition represents the action of $\Lambda$ on the space of Lorentz vectors $\text{Vector}_d$ as a continuous linear map from $\text{Vector}_d$ to itself, which sends a vector $v$ to the transformed vector $\Lambda \cdot v$.

For any spatial dimension $d \in \mathbb{N}$, any Lorentz transformation $\Lambda$ in the Lorentz group, and any Lorentz vector $p \in \text{Vector}_d$, the application of the continuous linear map $\text{actionCLM}(\Lambda)$ to the vector $p$ is equal to the group action of $\Lambda$ on $p$, denoted as $\Lambda \cdot p$.

For any spatial dimension $d \in \mathbb{N}$ and any Lorentz transformation $\Lambda$ in the Lorentz group, the continuous linear map $\text{actionCLM } \Lambda : \text{Vector}_d \to \text{Vector}_d$ (which maps a vector $v$ to $\Lambda \cdot v$) is injective.

For any spatial dimension $d$ and any Lorentz transformation $\Lambda$ in the Lorentz group, the continuous linear map $\text{actionCLM } \Lambda: \text{Vector}_d \to \text{Vector}_d$, which maps a Lorentz vector $v$ to its transformed version $\Lambda \cdot v$, is surjective.

For a given spatial dimension $d$, let $\Lambda$ be a Lorentz transformation and $e_\mu$ denote the $\mu$-th basis vector of the standard basis for the space of Lorentz vectors $\text{Vector}_d$, where the index $\mu \in \text{Fin } 1 \oplus \text{Fin } d$. The action of the Lorentz transformation on this basis vector is given by the linear combination: $$\Lambda \cdot e_\mu = \sum_{\nu} \Lambda_{\nu\mu} e_\nu$$ where $\Lambda_{\nu\mu}$ represents the $(\nu, \mu)$-th entry of the matrix representing the Lorentz transformation $\Lambda$.