Dagger geometry as Banach algebraic geometry

Introductory Geometry: Algebraic Geometry

Algebraic Geometry

Physical objects and constraints may be modeled by polynomial equations and inequalities. For this reason algebraic geometry, the study of solutions to systems of polynomial equations, is a tool for scientists and engineers. Moreover, relations between concepts arising in science and engineering are often described by polynomials. Whatever their source, once polynomials enter the picture, notio...

Algebraic Geometry

Algebraic Geometry

Examples • If S = ∅, then Z(S) = An. • If S = {1}, then Z(S) = ∅. • If S = {x1 − a1, . . . , xn − an}, then Z(S) = {(a1, . . . , an)}. Remark. All rings in this course are commutative and have 1. Remark. If A is a ring, then any subset S ⊆ A generates a minimal ideal 〈S〉 ⊆ A. In fact, we have 〈S〉 = {∑ aj xj : aj ∈ A, xj ∈ S}. Lemma. Z(S) = Z(〈S〉) for all S ⊆ k[x1 , . . . , xn]. Proof. Since 〈S〉...

Some Banach Space Geometry

Notation. ε is an arbitrarily small positive number. L(X,Y ) is the Banach space of bounded linear operators T : X → Y . B(X) is the closed unit ball {x ∈ X : ‖x‖ 6 1}. N := N ∪ {∞} is the one-point compactification of the discrete space N. All operators are bounded and linear. A compact (topological) space always means a compact Hausdorff space. (We sometimes add “Hausdorff” explicitly for emp...