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Unit spheres in finite-dimensional normed spaces are sequentially compact

Statement

Let $V$ be a finite-dimensional real normed space with norm $∥ \cdot ∥$ and unit sphere $S = {v \in V : ∥ v ∥ = 1} .$ Then $S$ is sequentially compact: every sequence $(v_{k})$ in $S$ has a subsequence $(v_{k_{j}})$ converging in the norm metric of $V$ to some $v \in S$ .

Consequently, if $(M, g)$ is a finite-dimensional Riemannian manifold, $p \in M$ and $S_{p} M = {v \in T_{p} M : ∣ v ∣_{g} = 1},$ then $S_{p} M$ is sequentially compact: every sequence of unit tangent vectors at $p$ has a subsequence converging in the norm metric of $T_{p} M$ to a unit tangent vector at $p$ .

No choice principle is used. The empty sphere (dimension zero) and the two-point sphere (dimension one) are included.

Facts & Assumptions

Given: The finite-dimensional real normed space $(V, ∥ \cdot ∥)$ with unit sphere $S$ , and the finite-dimensional Riemannian manifold $(M, g)$ with point $p \in M$ for the consequence.

[F1]

A normed space carries the metric $d_{N} (u, v) = N (u - v)$ , and every metric notion — convergence, compactness, the subspace metric — is available in it with no second definition (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ ).

[F2]

The closed unit ball ${\overline{B}}_{X} = {x \in X : ∥ x ∥ \leq 1}$ of a normed space $X$ is compact in the norm metric if and only if $X$ admits an ordered basis of finite length (The closed unit ball is compact if and only if the normed space is finite-dimensional).

[F3]

A finite-dimensional $V$ has a basis $B$ with $B \approx n$ for some $n \in N$ ; an equinumerosity $B \approx n$ is a bijection $n \to B$ , and an ordered basis is a finite injective list whose image is a basis (Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis, Equinumerous sets, $A \approx B$ and $A ⪯ B$ , Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis).

[F4]

A compact metric space is countably compact, and a countably compact metric space is sequentially compact; both implications are theorems of ZF (In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle).

[F5]

Sequentially compact means that every sequence has a subsequence converging to a point of the space (Countably compact, sequentially compact and limit point compact metric spaces, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ ).

[F6]

For all $x, y$ in a normed space, $∣ ∥ x ∥ - ∥ y ∥ ∣ \leq ∥ x - y ∥$ (The reverse triangle inequality in a normed space).

[F7]

For $A \subseteq X$ the subspace metric is the restriction $d_{A} = d_{X} ↾ (A \times A)$ , so convergence in $(A, d_{A})$ is convergence in $(X, d_{X})$ with the limit in $A$ (Isometry, isometric embedding, and the subspace metric on a subset, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ ).

[F8]

For a Riemannian metric $g$ , each $g_{p}$ is a positive definite symmetric bilinear form on $T_{p} M$ ; the pointwise norm is $∣ v ∣_{g} = \sqrt{g (v, v)}$ , and the length induced by an inner product is a norm (Riemannian metric and riemannian manifold, Pointwise norm and angle from a riemannian metric, The induced length is a norm).

[F9]

If $M$ is a smooth $n$ -manifold and $p \in M$ , then $T_{p} M$ is an $n$ -dimensional real vector space (The tangent space of an n-manifold has dimension n).

Proof

Proof technique: the closed unit ball is compact by finite dimensionality; its compactness passes to subsequences of the sphere, and the limit stays on the sphere because the norm is distance-controlling.

1.1

${\overline{B}}_{V} = {v \in V : ∥ v ∥ \leq 1}$ is compact. [F2, F3] Since $V$ is finite-dimensional, [F3] supplies a basis $B$ with $B \approx n$ for some $n \in N$ ; the bijection $n \to B$ witnessing the equinumerosity is an injective list whose image is a basis of $V$ , that is, an ordered basis of finite length $n$ . The implication from clause 2 to clause 1 of [F2] therefore gives compactness of ${\overline{B}}_{V}$ in the norm metric.

2.1

Let $(v_{k})$ be a sequence in $S$ . [F4, F5, F1, step 1.1] Then $∥ v_{k} ∥ = 1 \leq 1$ , so $(v_{k})$ is a sequence in ${\overline{B}}_{V}$ . By the first implication of [F4] the compact space ${\overline{B}}_{V}$ is countably compact, and by the second it is sequentially compact; hence there are a strictly increasing index map $k$ and a point $w \in {\overline{B}}_{V}$ with $v_{k_{j}} \to w$ in the norm metric of $V$ . Sequential compactness of the metric space ${\overline{B}}_{V}$ is applied to the sequence $(v_{k})$ read in that subspace; by [F7] the resulting convergence is convergence in $V$ to the same limit $w$ , which lies in ${\overline{B}}_{V}$ .

3.1

$w \in S$ . [F6, step 2.1] For every $j$ we have $∥ v_{k_{j}} ∥ = 1$ , and [F6] gives $∣ ∥ v_{k_{j}} ∥ - ∥ w ∥ ∣ \leq ∥ v_{k_{j}} - w ∥$ . Since $v_{k_{j}} \to w$ in the norm metric $d (u, v) = ∥ u - v ∥$ , the right-hand side tends to $0$ , so $∥ w ∥ = 1$ and $w \in S$ ; moreover $v_{k_{j}} \to w$ with $w \in S$ is convergence in the subspace $(S, d_{S})$ by [F7].

4.1

$S$ is sequentially compact. [F5, F7, step 2.1, step 3.1] Steps 2.1 and 3.1 take an arbitrary sequence in $S$ to a subsequence converging in the norm metric to a point of $S$ , which by [F7] is convergence in the subspace metric $d_{S}$ ; that is exactly the condition defining sequential compactness in [F5].

5.1

The Riemannian consequence holds. [F8, F9, step 4.1] By [F8], $g_{p}$ is a positive definite symmetric bilinear form on the real vector space $T_{p} M$ , and $∣ v ∣_{g} = \sqrt{g (v, v)}$ is the norm it induces; by [F9], $T_{p} M$ is $n$ -dimensional for $n = \dim M$ , hence finite-dimensional. So $V = T_{p} M$ with $∥ \cdot ∥ = ∣ \cdot ∣_{g}$ is a finite-dimensional real normed space and $S_{p} M$ is its unit sphere, so step 4.1 applies verbatim.

6.1

Boundary and choice audit. [F2, F4, F8, step 1.1, step 4.1, step 5.1] In dimension zero, $V = {0}$ and $S = \emptyset$ : the closed unit ball is the singleton ${0}$ , compact by step 1.1 with the empty ordered basis, and the sequential-compactness condition holds vacuously because there is no sequence into $S$ . In dimension one, $S = {\pm u}$ for a unit vector $u$ and compactness is immediate. The empty manifold carries no point $p$ , so the Riemannian consequence has no instance there. Finally, no choice principle is used: the implication of [F2] that is applied is the direction from a finite ordered basis to compactness of the ball, both implications of [F4] are theorems of ZF, and the limit computation of step 3.1 uses only the inequality of [F6]. $□$

Source locator

The closed unit ball of a finite-dimensional normed space is compact; this is the supplied library form of the Teschl argument, and the resulting sequential compactness of the unit sphere is the prerequisite used by Datar, Lectures on Riemannian Geometry, Lemma 23.2.2 proof, printed pp.167-168, where the limiting-direction argument passes to a convergent subsequence of unit tangent vectors without proving compactness. The tangent-space dimension is the supplied smooth-manifold corollary. The derivation above combines these library items and is carried out rather than quoted.

Depends on

Used by

Characterization of a cut point Theorem
Cut time is positive and continuous Theorem
The cut locus of a point is closed Theorem

Dependency tree · two levels

82 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (2025) (standard reference, not scraped)