Coordinate derivatives on real tensors #
The tensor representation already carries its Euclidean norm. Reading its coordinates is a continuous linear equivalence with a finite function space, so coordinatewise derivative proofs give Fréchet derivatives on the tensor itself. In particular, a matrix proof can keep its row and column indices instead of introducing a second flattened layout.
The coordinate function space is only used to assemble the derivative. The result has the native tensor type and the Euclidean topology used by the tensor inner-product and adjoint theorems.
The native coordinate view, with continuity supplied by finite dimensionality.
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Reading the coordinate view performs the same lookup as the tensor representation.
Rebuilding a tensor preserves each supplied coordinate.
A single tensor coordinate is a continuous linear functional.
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The coordinate functional uses the native lookup.
Local smoothness of a tensor is equivalent to local smoothness of every coordinate.
Reading a fixed coordinate preserves local smoothness.
An elementwise map needs smoothness only at the current coordinate values.
Locally smooth coordinate families assemble into a locally smooth native tensor.
A binary elementwise map needs smoothness only at each current pair of coordinates.
Tensor smoothness is equivalent to smoothness of its finitely many coordinates.
Reading a fixed tensor coordinate preserves the differentiability order.
Mapping a smooth scalar function over a smooth tensor preserves its differentiability order.
Smooth coordinate families assemble into a smooth native tensor.
Reading a packed storage index preserves local smoothness.
Packed row-major lookups are the same continuous coordinate functionals.
Summation preserves local smoothness, including for empty tensors.
Summing the packed tensor preserves smoothness, including when it has no entries.
Applying a smooth binary scalar function to two tensors preserves their smoothness.
Dot products preserve local smoothness in both tensor arguments.
Dot products of two smooth tensor families are smooth.
Matrix multiplication needs smoothness of its inputs only near the evaluation point.
Matrix multiplication preserves smoothness when both operands vary.
Fixed coordinate reads preserve local smoothness, including repeated reads and empty outputs.
Fixed coordinate reads preserve smoothness, including repeated reads and empty outputs.
Assemble one scalar derivative for each output coordinate into a tensor derivative.
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Each coordinate of the assembled tangent is given by its supplied scalar derivative.
Scalar coordinate derivatives determine the full tensor Fréchet derivative.
Finiteness of the shape matters here: assembling the coordinates is a continuous linear map. The hypotheses therefore imply a Fréchet derivative in the tensor norm, not just separate directional derivatives of its entries.