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Does the Banach algebra of jets have the approximation property?
Is $x_0+\tau x_0' \mapsto (x_0'(\cdot), x_0(0))$ a linear homeomorphism from $J^1$ to $C([0,1])\times \mathbb R$? Wouldn't that imply that $J^1$ inherits the approximation property from $C^0([0,1])$?