Proof that finite-dimensional normed spaces are complete

Builds on Banach space Complete metric space Epsilon characterisation of Cauchy sequences Equivalence of norms in finite dimension Equivalent norms Finite-dimensional vector space Every spanning set contains a basis Isomorphism with the coordinate space Completeness of the real numbers Convergence of a sequence in a metric space

Proof

We compare the norm with the sup norm of the coordinates and use completeness of $\mathbb{R}$ in each coordinate.

Let $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$ . If $V = \{0\}$ , each sequence is constant and converges, so $V$ is complete. Let $V \neq \{0\}$ .
Since $V$ is finite-dimensional, it has a basis $e_1, \ldots, e_n$ . Write $x = \sum_j a_j(x) e_j$ , with linear coordinates $a_j$ , by this result.
Let $\|x\|_\infty = \max_j |a_j(x)|$ . It is a norm: it vanishes only when all coordinates vanish, it is homogeneous, and $|a_j(x + y)| \le |a_j(x)| + |a_j(y)|$ . By the equivalence of norms in finite dimension, there are $c, C > 0$ with $c \|x\|_\infty \le \|x\| \le C \|x\|_\infty$ for all $x$ , by the definition of equivalent norms.
Let $(x_k)$ be a Cauchy sequence in $(V, \|\cdot\|)$ . Fix $j$ . Then $|a_j(x_k) - a_j(x_l)| \le \|x_k - x_l\|_\infty \le \|x_k - x_l\| / c$ . So $(a_j(x_k))_k$ is a Cauchy sequence in $\mathbb{K}$ .
Its real parts and imaginary parts form Cauchy sequences in $\mathbb{R}$ , since $|\operatorname{Re} z| \le |z|$ and $|\operatorname{Im} z| \le |z|$ .
By the completeness of the real numbers, both converge. So $a_j(x_k) \to \alpha_j$ for some $\alpha_j \in \mathbb{K}$ .
Let $x = \sum_j \alpha_j e_j$ . Then $a_j(x_k - x) = a_j(x_k) - \alpha_j$ , so $\|x_k - x\| \le C \max_j |a_j(x_k) - \alpha_j|$ .
Given $\varepsilon > 0$ , for each $j$ there is $N_j$ with $|a_j(x_k) - \alpha_j| < \varepsilon / C$ for $k \ge N_j$ . For $k \ge \max_j N_j$ , we get $\|x_k - x\| < \varepsilon$ .
So $(x_k)$ converges to $x$ . Each Cauchy sequence converges, so $V$ is complete, and $V$ is a Banach space.

Idea

All norms on a finite-dimensional space are equivalent, so we may measure with the largest coordinate. Then a Cauchy sequence of vectors gives finitely many Cauchy sequences of numbers, which converge because $\mathbb{R}$ is complete.

Reading path

To understand this page from the ground up, read these 202 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 the triangle inequality for norms, Proof of the Heine–Borel theorem by sequential compactness, 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 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 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 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 a continuous real function on a compact set attains its bounds, Proof that a decreasing sequence bounded below converges, Proof of the monotone convergence theorem for sequences, Proof that open balls are open, Proof that a set is open if and only if its complement is closed, 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
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
Addition of natural numbers definition
Existential quantifier definition
Universal quantifier definition
The successor function is injective statement
Zero is not a successor statement
Bounded sequence definition
Cauchy sequence definition
Limit of a sequence definition
Monotone sequence definition
The rationals form a field statement
Linear independence definition
Linear isomorphism definition
Linear span definition
Completeness axiom statement
Cartesian product definition
Union of a family of sets definition
Subset definition
Uniqueness quantifier definition
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
Convergent sequence definition
The rationals form an ordered field statement
Spanning set definition
Binary relation definition
Topological space definition
Universal set definition
Order on the reals 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
Function definition
Subspace topology definition
Complement definition
Coordinate vector definition
Strong induction statement
Preorder definition
Choice function definition
Image of a set definition
Preimage of a set 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
Partial order definition
Axiom of choice statement
Surjective function definition
Left inverse definition
Right inverse definition
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
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
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
Open cover definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
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
Compact space definition
Open sets characterised by neighbourhoods statement
Proof that a set is open if and only if its complement is closed proof
Equivalent norms definition
Proof that the coordinate map is a linear isomorphism proof
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
Complement of an open set is closed statement
Compact set definition
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
Closed sets contain their limits statement
Open balls are open statement
Epsilon-N definition of the limit of a sequence statement
Square root definition
Compactness via open covers statement
Modulus of a complex number definition
Continuity via closed sets statement
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 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
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
Epsilon-delta continuity in metric spaces statement
Lebesgue number lemma statement
Proof that compact subsets of a metric space are bounded proof
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
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
Continuous image of a compact set is compact statement
Compactness and sequential compactness agree in metric spaces statement
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
Proof that a decreasing sequence bounded below converges proof
Triangle inequality for norms statement
Decreasing sequence bounded below converges statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Bolzano–Weierstrass theorem statement
Proof of the Heine–Borel theorem by sequential compactness proof
Completeness of the real numbers statement
Heine–Borel theorem 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

Where this leads

226 pages build on this one.

Tag	`RG08WK1WDK`
URI	`m2://RG08WK1WDK`
Kind	proof
Writing	W0 stub
MSC	46B99
Proves	Finite-dimensional normed spaces are complete
Uses	Banach space, Complete metric space, Epsilon characterisation of Cauchy sequences, Equivalence of norms in finite dimension, Equivalent norms, Finite-dimensional vector space, Every spanning set contains a basis, Isomorphism with the coordinate space, Completeness of the real numbers, Convergence of a sequence in a metric space
Changed	4 November 2026

Proof that finite-dimensional normed spaces are complete

Proof

Idea

Reading path

Where this leads

Further reading

What this page needs for W1

What links here