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  "body": "\n## Definition\n\nLet $M$ be a [[Banach manifold|VB95VVVDM8]] of class $C^p$, with $p \\ge 0$.\nLet $E$ be a topological space and $\\pi \\colon E \\to M$ a continuous surjection, and assume each fibre $E_x = \\pi^{-1}(x)$ carries the structure of a real [[Banach space|0BDTTM24DP]].\nFor Banach spaces $X$ and $Y$, let $\\mathcal{L}(X; Y)$ be the Banach space of [[bounded linear maps|0S3SMRH0Z5]] with the [[operator norm|0ZTEH055YF]].\nThe map $\\pi$ is a *Banach bundle* of class $C^p$ if there are data of the following kind.\nThere are an open cover $(U_i)_{i \\in I}$ of $M$, Banach spaces $X_i$ and maps $\\tau_i \\colon \\pi^{-1}(U_i) \\to U_i \\times X_i$ such that:\n\n\n1. Each $\\tau_i$ is a [[homeomorphism|0QT00PB5G2]] with $\\mathrm{pr}_{U_i} \\circ \\tau_i = \\pi$.\n2. For each $x \\in U_i$, the induced map $\\tau_{i,x} \\colon E_x \\to X_i$ is a continuous linear isomorphism.\n3. For all $i, j \\in I$, the map $U_i \\cap U_j \\to \\mathcal{L}(X_i; X_j)$, $x \\mapsto \\tau_{j,x} \\circ \\tau_{i,x}^{-1}$, is of class $C^p$.\n\n## Remarks\n\nExample: for a Banach manifold $M$ and a Banach space $X$, the product $M \\times X \\to M$ with the single trivialisation $\\tau = \\id$ is a Banach bundle; the transition map is constant.\nThe tangent bundle of a $C^{p+1}$ Banach manifold is a Banach bundle of class $C^p$, with transition maps given by derivatives of chart changes.\n\nNon-example: clause 3 asks for continuity in the operator norm, not only pointwise.\nOn $X = L^2(\\R)$ let $T_s f(t) = f(t - s)$ be the translation by $s$.\nThen $s \\mapsto T_s f$ is continuous for each $f$, but $\\norm{T_s - T_{s'}} = 2$ for $s \\ne s'$.\nSo gluing two trivial bundles $(-\\infty, 1) \\times X$ and $(0, \\infty) \\times X$ over $M = \\R$ by $x \\mapsto T_{1/x}$ on $(0, 1)$ violates clause 3.\n\nWhen all $X_i$ are finite-dimensional, this is the notion of a vector bundle of class $C^p$.\n",
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