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topology.continuous_function.polynomial

Constructions relating polynomial functions and continuous functions. #

Main definitions #

Every polynomial with coefficients in a topological semiring gives a (bundled) continuous function.

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A polynomial as a continuous function, with domain restricted to some subset of the semiring of coefficients.

(This is particularly useful when restricting to compact sets, e.g. [0,1].)

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@[simp]
theorem polynomial.aeval_continuous_map_apply {R : Type u_1} {α : Type u_2} [topological_space α] [comm_semiring R] [topological_space R] [topological_ring R] (g : polynomial R) (f : C(α, R)) (x : α) :

The algebra map from polynomial R to continuous functions C(R, R).

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The algebra map from polynomial R to continuous functions C(X, R), for any subset X of R.

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The subalgebra of polynomial functions in C(X, R), for X a subset of some topological ring R.

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The preimage of polynomials on [0,1] under the pullback map by x ↦ (b-a) * x + a is the polynomials on [a,b].