Proof that a compact closed unit ball implies finite dimension

Builds on Riesz's lemma Compactness and sequential compactness agree in metric spaces Sequential compactness Finite-dimensional normed spaces are complete Complete subspaces are closed Finite-dimensional vector space Linear span Linear subspace Convergence of a sequence in a metric space

Proof

We argue by contradiction: in infinite dimension, Riesz's lemma gives unit vectors that stay far apart.

Let $\bar{B} = \{x \in X : \|x\| \le 1\}$ be compact, and suppose that $X$ is not finite-dimensional. Then $X$ has no finite spanning set. In particular $X \neq \{0\}$ .
We build unit vectors $x_1, x_2, \ldots$ with $\|x_m - x_n\| \ge \tfrac{1}{2}$ for $m \neq n$ . Choose any $x_1 \neq 0$ and replace it by $x_1 / \|x_1\|$ .
Given $x_1, \ldots, x_n$ , let $Y_n$ be their linear span. It is a linear subspace, and $Y_n \neq X$ because $X$ has no finite spanning set.
$Y_n$ with the restricted norm is a finite-dimensional normed space. So it is complete, by finite-dimensional normed spaces are complete.
Its metric is the restriction of the metric of $X$ . So $Y_n$ is closed in $X$ , by complete subspaces are closed.
By Riesz's lemma with $\alpha = \tfrac{1}{2}$ , there is $x_{n+1}$ with $\|x_{n+1}\| = 1$ and $\|x_{n+1} - y\| \ge \tfrac{1}{2}$ for all $y \in Y_n$ .
For $m > n$ , we have $x_n \in Y_{m-1}$ . So $\|x_m - x_n\| \ge \tfrac{1}{2}$ , by step 6 for $x_m$ .
All $x_n$ lie in $\bar{B}$ . As $\bar{B}$ is compact in the metric space $X$ , it is sequentially compact, by this result.
So some subsequence $(x_{n_k})$ converges to a point $z$ . There is $k_0$ with $\|x_{n_k} - z\| < \tfrac{1}{4}$ for $k \ge k_0$ .
Then $\|x_{n_{k_0}} - x_{n_{k_0 + 1}}\| \le \|x_{n_{k_0}} - z\| + \|z - x_{n_{k_0 + 1}}\| < \tfrac{1}{2}$ , with $n_{k_0} \neq n_{k_0 + 1}$ . This contradicts step 7.
So $X$ is finite-dimensional.

Idea

In infinite dimension, each finite-dimensional subspace is a closed proper subspace. Riesz's lemma then gives a new unit vector far from all previous ones. The resulting sequence has no convergent subsequence, so the unit ball cannot be compact.

Reading path

To understand this page from the ground up, read these 239 pages in order. The foundations come first. The path takes the proofs Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the diameter characterisation of Cauchy sequences, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of Riesz's lemma, Proof of the triangle inequality for norms, Proof of De Morgan's laws for sets, Proof of De Morgan's laws by truth tables, Proof of the Heine–Borel theorem by sequential compactness, Proof that a union of open sets of a metric space is open, Proof that an intersection of closed sets of a metric space is closed, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of continuity via closed sets, Proof that complete subspaces are closed, Proof that compact subsets of a metric space are bounded, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof of the existence of square roots of nonnegative reals, Proof that a continuous image of a compact set is compact, Proof that the closure is the smallest closed superset, Proof that compact subsets of a Hausdorff space are closed, Proof of epsilon-delta continuity at a point in metric spaces, Proof of the description of the open sets of a metric space, Proof of the epsilon-N definition of the limit of a sequence, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof of the description of the closure in a metric space, Proof that every spanning set contains a basis, by Zorn's lemma, Proof that closed sets contain their limits, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof that finite-dimensional normed spaces are complete, Proof that a continuous real function on a compact set attains its bounds, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof of the monotone convergence theorem for sequences, Proof that open balls are open, Proof that the closure is closed, Proof that a set is open if and only if its complement is closed, Proof that a set is closed if and only if it equals its closure, Proof that the closure is the set of limits of sequences, Proof of continuity via open sets in metric spaces.

Basis concept
Complex numbers object
Construction of the reals from Cauchy sequences concept
Coordinate space object
Coordinate vector concept
Definition by recursion concept
Element definition
Euclidean metric example
Extended real number line object
Integers object
Linear isomorphism concept
Linear map concept
Mathematical induction concept
Natural numbers object
Norm concept
Ordered field definition
Peano axioms concept
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Vector space definition
p-metric on R^n example
Real and imaginary parts definition
Set definition
Ring of integers statement
Successor function definition
Archimedean ordered field definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Linear combination definition
Linear map definition
Linear subspace definition
Norm definition
p-norm on R^n example
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Biconditional definition
Truth table concept
Addition of natural numbers definition
Existential quantifier definition
Universal quantifier definition
The successor function is injective statement
Zero is not a successor statement
The rationals form a field statement
Bounded sequence definition
Cauchy sequence definition
Limit of a sequence definition
Monotone sequence definition
Linear independence definition
Linear isomorphism definition
Linear span definition
Completeness axiom statement
Cartesian product definition
Intersection of a family of sets definition
Union of a family of sets definition
Axiom of extensionality statement
Subset definition
Proof of the commutativity of addition by induction proof
Propositional logic concept
Uniqueness quantifier definition
Order on the natural numbers definition
The rationals form an ordered field statement
Convergent sequence definition
Spanning set definition
Binary relation definition
Topological space definition
Universal set definition
Order on the reals definition
Tautology definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Finite-dimensional vector space definition
Basis definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Sequential compactness definition
Subspace topology definition
Function definition
Complement definition
Logical equivalence definition
Coordinate vector definition
Strong induction statement
Preorder definition
Choice function definition
Image of a set definition
Preimage of a set definition
Restriction of a function definition
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Proof of the well-ordering principle from strong induction proof
Metric space definition
Proof of De Morgan's laws by truth tables proof
Partial order definition
Axiom of choice statement
Surjective function definition
Subspace metric definition
Left inverse definition
Right inverse definition
De Morgan's laws statement
Well-ordering principle statement
Closed ball definition
Convergence of a sequence in a metric space definition
Lipschitz map definition
Open ball definition
Triangle inequality for a metric statement
Diameter of a set definition
Maximal element definition
Total order definition
Upper bound definition
Proof of De Morgan's laws for sets proof
Bijective function definition
Metric topology definition
Bounded set definition
Proof that every sequence has a monotone subsequence, by peaks proof
Proof of the diameter characterisation of Cauchy sequences proof
Chain definition
Open set definition
De Morgan's laws for sets statement
Proof that a function is bijective iff it has a two-sided inverse proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Epsilon characterisation of Cauchy sequences statement
Every sequence has a monotone subsequence statement
Hausdorff space definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Open cover definition
Continuous map of topological spaces definition
Closed set definition
Bijective iff it has a two-sided inverse statement
Zorn's lemma statement
Equivalent metrics definition
Proof of the description of the open sets of a metric space proof
Complete metric space definition
The reals form an ordered field statement
Open sets characterised by neighbourhoods statement
Closure definition
Compact space definition
Proof that a set is open if and only if its complement is closed proof
Proof that the coordinate map is a linear isomorphism proof
Equivalent norms definition
Open sets of a metric space statement
Proof that every spanning set contains a basis, by Zorn's lemma proof
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Proof of the existence of square roots of nonnegative reals proof
Continuity at a point of a topological space definition
Proof that the closure is closed proof
Complement of an open set is closed statement
Compact set definition
Proof that a union of open sets of a metric space is open proof
Banach space definition
Proof that closed sets contain their limits proof
Every spanning set contains a basis statement
Proof that open balls are open proof
Isomorphism with the coordinate space statement
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Existence of the infimum statement
Existence of square roots of nonnegative reals statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Proof of continuity via closed sets proof
Union of open sets is open statement
Closed sets contain their limits statement
Open balls are open statement
Epsilon-N definition of the limit of a sequence statement
Square root definition
Proof that the reals are Archimedean from the completeness axiom proof
Compactness via open covers statement
Modulus of a complex number definition
Continuity via closed sets statement
Proof that an intersection of closed sets of a metric space is closed proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of continuity via open sets in metric spaces proof
Proof of the description of the closure in a metric space proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof that metric spaces are Hausdorff proof
Proof that subsequences of a convergent sequence converge to the same limit proof
The reals are Archimedean statement
Proof of the monotone convergence theorem for sequences proof
Proof that compact subsets of a Hausdorff space are closed proof
Proof that a continuous image of a compact set is compact proof
Intersection of closed sets is closed statement
Epsilon-delta continuity in metric spaces statement
Closure in a metric space statement
Lebesgue number lemma statement
Proof that compact subsets of a metric space are bounded proof
Proof that compactness and sequential compactness agree in metric spaces proof
Metric spaces are Hausdorff statement
Proof of the triangle inequality for norms proof
Subsequences of a convergent sequence converge to the same limit statement
Monotone convergence theorem for sequences statement
Compact subsets of a Hausdorff space are closed statement
Closure is closed statement
Continuous image of a compact set is compact statement
Continuity via open sets statement
Compactness and sequential compactness agree in metric spaces statement
Compact subsets of a metric space are bounded statement
Proof that a decreasing sequence bounded below converges proof
Triangle inequality for norms statement
Proof that the closure is the smallest closed superset proof
Decreasing sequence bounded below converges statement
Closure is the smallest closed superset statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof that a set is closed if and only if it equals its closure proof
Proof that the closure is the set of limits of sequences proof
Bolzano–Weierstrass theorem statement
A set is closed iff it equals its closure statement
Closure is the set of limits of sequences statement
Proof of the Heine–Borel theorem by sequential compactness proof
Completeness of the real numbers statement
Heine–Borel theorem statement
Proof that complete subspaces are closed proof
Proof of Riesz's lemma proof
Complete subspaces are closed statement
Riesz's lemma statement
Proof that a continuous real function on a compact set attains its bounds proof
Continuous real function on a compact set attains its bounds statement
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Equivalence of norms in finite dimension statement
Proof that finite-dimensional normed spaces are complete proof
Finite-dimensional normed spaces are complete statement
Proof that a compact closed unit ball implies finite dimension proof

Where this leads

1 page builds on this one.

Compact closed unit ball implies finite dimension

Tag	`G96C77E433`
URI	`m2://G96C77E433`
Kind	proof
Writing	W0 stub
MSC	46B50
Proves	Compact closed unit ball implies finite dimension
Uses	Riesz's lemma, Compactness and sequential compactness agree in metric spaces, Sequential compactness, Finite-dimensional normed spaces are complete, Complete subspaces are closed, Finite-dimensional vector space, Linear span, Linear subspace, Convergence of a sequence in a metric space
Changed	4 November 2026

Proof that a compact closed unit ball implies finite dimension

Proof

Idea

Reading path

Where this leads

Further reading

What this page needs for W1

What links here