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In the book "a convenient setting for global analysis" they describe the order of an operational tangent vector on a convenient vector space.

http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf

After which they prove that their exists those of order 2 and 3. After breaking my head on it for a bit I can't seem to wrap my head around the proof.

What is the second order vector they prove that exists in that case? I would already be quite happy with a concrete example.

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Let $\alpha\in B(H)'$ be a bounded linear functional on the space of all bounded linear operators on Hilbert space which vanishes on the subspace of compact operators $K(H)\supset H\otimes H'$. Then $f\mapsto \alpha(d^2f(0))$ for $f\in C^\infty(H)$ is an operational tangent vector with this property, since it is a derivation: \begin{align} \alpha(d^2(f.g)(0) &= \alpha\Big(d^2f(0).g(0) + df(0)\otimes dg(0) + dg(0)\otimes df(0) + f(0)d^2g(0)\Big) \\& = \alpha\big(d^2f(0)\big).g(0) + 0 + 0 + f(0).\alpha\big(d^2g(0)\big) \end{align} So convince yourself that these linear functionals exist, by Hahn-Banach.

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