> [!definitionb] Definition > > Let $E$ be a [[Banach Space|Banach space]], $X$ be an $E$-[[Manifold|manifold]], $p \in X$, and $(U_i, \psi_i)_{i \in I}$ be charts with $p \in U_i$ for all $i \in I$, then for any $i, j \in I$, the [[Derivative|derivative]] $D(\psi_j \circ \psi_{i}^{-1})(p) \in \text{Laut}(E)$ is a [[Space of Toplinear Isomorphisms|toplinear automorphism]], and the [[Concrete Tangent Space|concrete tangent spaces]] $T_{(U_i, \psi_i, p)}X$ and $T_{U_j, \psi_j, p}$ are toplinearly isomorphic. Let > $ > \widehat{T_pX} = \bigcup_{i \in I}T_{(U_i, \psi_i, p)} > $ > be the union of all concrete tangent vectors at $p$, and let $\widehat{\topo}$ be the [[Topological Space|topology]] generated by the union of the topologies on each concrete tangent space. > > Define the [[Concrete Tangent Space|concrete tangent vectors]] > $ > (U_i, \psi_i, p, v) \sim (U_j, \psi_j, p, w) \Leftrightarrow w = D(\psi_2 \circ \psi_{1}^{-1})(p)(v) > $ > to be equivalent if one is the image of another by the above automorphism, and each equivalence class to be an *abstract* **tangent vector**. > > Let $T_pX$ be the collection of all such [[Equivalence Class|equivalence classes]], then $T_pX$ is the **tangent space** at $p$. Let $\topo$ be the [[Quotient Topology|quotient topology]] obtained from $\widehat\topo$, then $T_pX$ is [[Space of Toplinear Isomorphisms|toplinearly isomorphic]][^1] to $T$. > > *Remark: While it is possible to assign a norm to the tangent space, it is not very meaningful as each chart's norm would disagree with it. However, the topological structure is still preserved.* > [!definition] > > Let $V$ be a [[Vector Space|vector space]] and $p \in V$, then the collection > $ > V_p = \bracs{(p, v): v \in V} > $ > is the **tangent space** of $V$ at $p$. Points in $V_p$ are denoted as $v_p$. > > The tangent space inherits structures on $V$ such as the [[Normed Vector Space|norm]], the [[Inner Product|inner product]], and [[Orientation|orientations]] by operating on the second coordinate. > [!definition] > > Let $V_p$ be a tangent space and $v_p \in V_p$, then $v + p$ is its **end point**. > [!definition] > > Let $X$ be a [[n-Manifold|n-manifold]], then $T_pM \iso \real^n$ for all $p \in X$. If $(U, \varphi)$ is a chart at $p$, then $\{\frac{\partial}{\partial x^i}\vert_p\}_1^n$ is a [[Basis|basis]] for $T_pX$, known as a **coordinate basis** for $T_pX$. > > *Proof*. The map $\varphi: U \to \widehat U$ is a [[Diffeomorphism|diffeomorphism]], so $d\varphi_p: T_pX \to T_{\hat p}\real^n$ is an isomorphism. Since $T_{\hat p}\real^n$ is [[Derivation on Rn|is spanned by the partial derivatives]], their preimages by $d\varphi_p$ is a basis consisting of the partial derivatives.