with_gaugeSeminormFamily

theorem with_gaugeSeminormFamily : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : AddCommGroup E] [inst_2 : TopologicalSpace E] [inst_3 : Module 𝕜 E] [inst_4 : Module ℝ E] [inst_5 : IsScalarTower ℝ 𝕜 E] [inst_6 : ContinuousSMul ℝ E] [inst_7 : TopologicalAddGroup E] [inst_8 : ContinuousSMul 𝕜 E] [inst_9 : SMulCommClass ℝ 𝕜 E] [inst_10 : LocallyConvexSpace ℝ E], WithSeminorms (gaugeSeminormFamily 𝕜 E)

The theorem `with_gaugeSeminormFamily` states that for any locally convex space over a real number field, its topology is induced by the family of gauge seminorms defined by the gauges of absolute convex open sets. Here, the types 𝕜 and E represent the real number field and the locally convex space respectively. The additional instances are conditions on E, such as it being a topological space, a module over 𝕜 and ℝ, and having continuous scalar multiplication. The theorem is part of the theory of locally convex vector spaces, which are foundational in functional analysis and related areas of mathematics.

More concisely: The topology of a locally convex space over the real number field is induced by the family of gauge seminorms of its absolute convex open sets.