A linear isomorphism is determined by its action on an ordered basis or "'frame " '.
2.
If one considers a Riemannian manifold or a pseudo-Riemannian manifold, the Riemannian metric induces a linear isomorphism between the tangent and cotangent bundles . ( See Musical isomorphism ).
3.
I think one other important thing to consider is that in manifold theory there are important facts about the derivative of a smooth function between smooth manifolds ( namely it is a linear isomorphism between tangent spaces ).
4.
A 1969 theorem of David Henderson states that every infinite-dimensional, embedded as an open subset of the infinite-dimensional, separable Hilbert space, " H " ( up to linear isomorphism, there is only one such space ).
5.
Thus parallel transport provides a way of moving tangent vectors along a curve using the affine connection to keep them " pointing in the same direction " in an intuitive sense, and this provides a linear isomorphism between the tangent spaces at the two ends of the curve.
6.
In finite dimensions, the two notions of dual cone are essentially the same because every finite dimensional linear functional is continuous, and every continuous linear functional in a inner product space induces a linear isomorphism ( nonsingular linear map ) from " V * " to " V ", and this isomorphism will take the dual cone given by the second definition, in " V * ", onto the one given by the first definition; see the Riesz representation theorem.
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