inner product on vector bundles in nLab
The concept of inner product on vector bundles is the evident generalization of that of inner product on vector spaces to vector bundles: a fiber-wise inner product of vector spaces.
Proof
Let {U i⊂X} i∈I\{U_i \subset X\}_{i \in I} be an open cover of XX over which E→XE \to X admits a local trivialization
{ϕ i:U i×ℝ n⟶≃E| U i} i∈I. \left\{ \phi_i \;\colon\; U_i \times \mathbb{R}^n \overset{\simeq}{\longrightarrow} E|_{U_i} \right\}_{i \in I} \,.
Since paracompact Hausdorff spaces equivalently admit subordinate partitions of unity we may choose a partition of unity
{f i:X→[0,1]} i∈I. \{f_i \;\colon\; X \to [0,1]\}_{i \in I} \,.
Write
⟨−,−⟩ ℝ n:ℝ n⊗ℝ n⟶ℝ \langle -,-\rangle_{\mathbb{R}^n} \;\colon\; \mathbb{R}^n \otimes \mathbb{R}^n \longrightarrow \mathbb{R}
for the standard inner product on ℝ n\mathbb{R}^n.
By the compact support of f if_i inside U i⊂XU_i \subset X, the functions
⟨−,−⟩ i : E| U i⊗ XE| U i ⟶ϕ i −1⊗ U iϕ i −1 (U i×ℝ n)⊗ U i(U i×ℝ n) →≃ U i×(ℝ n⊗ℝ n) ⟶ U i×ℝ ((x,i),(v 1,v 2)) ↦AAA ((x,i),f i(x)⋅⟨v 1,v 2⟩ ℝ n). \array{ \langle -,-\rangle_i &\colon& E|_{U_i} \otimes_X E|_{U_i} &\overset{ \phi_i^{-1} \otimes_{U_i} \phi_i^{-1} }{\longrightarrow}& (U_i \times \mathbb{R}^n ) \otimes_{U_i} (U_i \times \mathbb{R}^n) &\overset{\simeq}{\to}& U_i \times (\mathbb{R}^n \otimes \mathbb{R}^n) &\overset{}{\longrightarrow}& U_i \times \mathbb{R} \\ && && && ((x,i),(v_1,v_2)) & \overset{\phantom{AAA}}{\mapsto} & ((x,i), f_i(x) \cdot \langle v_1, v_2\rangle_{\mathbb{R}^n} ) } \,.
extend by zero to continuous functions on all of E⊗ XEE \otimes_X E, which we denote by the same symbol ⟨−,−⟩ i:E⊗ XE→X×ℝ\langle -,-\rangle_i \colon E \otimes_X E \to X \times \mathbb{R}.
We then claim that the sum
⟨−,−⟩:∑i∈I⟨−,−⟩ i:E⊗ XE⟶X×ℝ \langle -,-\rangle \;\colon\; \underset{i \in I}{\sum} \langle -,-\rangle_i \;\colon\; E \otimes_X E \longrightarrow X \times \mathbb{R}
is an inner product as required. Notice that the sum is well defined by local finiteness of the supports of the partition functions f if_i. Hence it is pointwise a finite sum of positive definite symmetic bilinear forms on E xE_x, and as such itself pointwise a positive definite symmetric bilinear form.