File: Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean

full_name: toEuclidean
code: /-- If `E` is a finite dimensional space over `ℝ`, then `toEuclidean` is a continuous `ℝ`-linear
equivalence between `E` and the Euclidean space of the same dimension. -/
def toEuclidean : E ≃L[ℝ] EuclideanSpace ℝ (Fin <| finrank ℝ E) 
goal: E ≃L[ℝ] EuclideanSpace ℝ (Fin (FiniteDimensional.finrank ℝ E))

full_name: Euclidean.dist
code: /-- If `x` and `y` are two points in a finite dimensional space over `ℝ`, then `Euclidean.dist x y`
is the distance between these points in the metric defined by some inner product space structure on
`E`. -/
nonrec def dist (x y : E) : ℝ 
goal: ℝ

full_name: Euclidean.closedBall
code: /-- Closed ball w.r.t. the euclidean distance. -/
def closedBall (x : E) (r : ℝ) : Set E 
goal: Set E

full_name: Euclidean.ball
code: /-- Open ball w.r.t. the euclidean distance. -/
def ball (x : E) (r : ℝ) : Set E 
goal: Set E

full_name: Euclidean.ball_eq_preimage
code: theorem ball_eq_preimage (x : E) (r : ℝ) :
    ball x r = toEuclidean ⁻¹' Metric.ball (toEuclidean x) r 
goal: Euclidean.ball x r = ⇑toEuclidean ⁻¹' Metric.ball (toEuclidean x) r

full_name: Euclidean.closedBall_eq_preimage
code: theorem closedBall_eq_preimage (x : E) (r : ℝ) :
    closedBall x r = toEuclidean ⁻¹' Metric.closedBall (toEuclidean x) r 
goal: Euclidean.closedBall x r = ⇑toEuclidean ⁻¹' Metric.closedBall (toEuclidean x) r

full_name: Euclidean.ball_subset_closedBall
code: theorem ball_subset_closedBall {x : E} {r : ℝ} : ball x r ⊆ closedBall x r 
goal: Euclidean.ball x r ⊆ Euclidean.closedBall x r

full_name: Euclidean.isOpen_ball
code: theorem isOpen_ball {x : E} {r : ℝ} : IsOpen (ball x r) 
goal: IsOpen (Euclidean.ball x r)

full_name: Euclidean.mem_ball_self
code: theorem mem_ball_self {x : E} {r : ℝ} (hr : 0 < r) : x ∈ ball x r 
goal: x ∈ Euclidean.ball x r

full_name: Euclidean.closedBall_eq_image
code: theorem closedBall_eq_image (x : E) (r : ℝ) :
    closedBall x r = toEuclidean.symm '' Metric.closedBall (toEuclidean x) r 
goal: Euclidean.closedBall x r = ⇑toEuclidean.symm '' Metric.closedBall (toEuclidean x) r

full_name: Euclidean.isCompact_closedBall
code: nonrec theorem isCompact_closedBall {x : E} {r : ℝ} : IsCompact (closedBall x r) 
goal: IsCompact (Euclidean.closedBall x r)

full_name: Euclidean.isClosed_closedBall
code: theorem isClosed_closedBall {x : E} {r : ℝ} : IsClosed (closedBall x r) 
goal: IsClosed (Euclidean.closedBall x r)

full_name: Euclidean.closure_ball
code: nonrec theorem closure_ball (x : E) {r : ℝ} (h : r ≠ 0) : closure (ball x r) = closedBall x r 
goal: closure (Euclidean.ball x r) = Euclidean.closedBall x r

full_name: Euclidean.exists_pos_lt_subset_ball
code: nonrec theorem exists_pos_lt_subset_ball {R : ℝ} {s : Set E} {x : E} (hR : 0 < R) (hs : IsClosed s)
    (h : s ⊆ ball x R) : ∃ r ∈ Ioo 0 R, s ⊆ ball x r 
goal: ∃ r ∈ Set.Ioo 0 R, s ⊆ Euclidean.ball x r

full_name: Euclidean.nhds_basis_closedBall
code: theorem nhds_basis_closedBall {x : E} : (𝓝 x).HasBasis (fun r : ℝ => 0 < r) (closedBall x) 
goal: (𝓝 x).HasBasis (fun r => 0 < r) (Euclidean.closedBall x)

full_name: Euclidean.closedBall_mem_nhds
code: theorem closedBall_mem_nhds {x : E} {r : ℝ} (hr : 0 < r) : closedBall x r ∈ 𝓝 x 
goal: Euclidean.closedBall x r ∈ 𝓝 x

full_name: Euclidean.nhds_basis_ball
code: theorem nhds_basis_ball {x : E} : (𝓝 x).HasBasis (fun r : ℝ => 0 < r) (ball x) 
goal: (𝓝 x).HasBasis (fun r => 0 < r) (Euclidean.ball x)

full_name: Euclidean.ball_mem_nhds
code: theorem ball_mem_nhds {x : E} {r : ℝ} (hr : 0 < r) : ball x r ∈ 𝓝 x 
goal: Euclidean.ball x r ∈ 𝓝 x

full_name: ContDiff.euclidean_dist
code: theorem ContDiff.euclidean_dist (hf : ContDiff ℝ n f) (hg : ContDiff ℝ n g) (h : ∀ x, f x ≠ g x) :
    ContDiff ℝ n fun x => Euclidean.dist (f x) (g x) 
goal: ContDiff ℝ n fun x => Euclidean.dist (f x) (g x)