Functional analysis begins when vectors stop being finite coordinate lists and become functions, sequences, signals or other infinite-dimensional objects—and when distance, convergence and linearity must still work reliably.
A normed space gives a notion of size. A Banach space adds completeness: every Cauchy sequence converges to a point already inside the space. That one property is the analytic foundation beneath fixed-point methods, differential and integral equations, optimisation, approximation theory and operator analysis.
This guide owns R27.01. It develops norms, convergence, completeness, standard sequence and function spaces, bounded linear maps, quotient and subspace structure, dual spaces and the major structural theorems that make Banach spaces usable. R27.02 adds inner products and orthogonal projection; R27.03 studies operators and spectra.
Prerequisites: Completeness of the Real Numbers, linear algebra and metric-space convergence.
Reading route: vector-space structure → norms → metric and convergence → Cauchy sequences → completeness → finite-dimensional equivalence → ℓp and function spaces → closed subspaces → bounded linear maps → operator norm → dual spaces → Hahn–Banach → Baire category → uniform boundedness → open mapping → closed graph → quotient spaces → counterexamples → practice and solutions.
1. A norm measures vector size without choosing coordinates
Let X be a vector space over R or C. A norm is a function ||·||:X→[0,∞) satisfying
- ||x||=0 iff x=0;
- ||αx||=|α| ||x||;
- ||x+y||≤||x||+||y||.
The pair (X,||·||) is a normed vector space.
2. Every norm creates a metric
Define
d(x,y)=||x−y||.
Translation invariance is automatic: d(x+a,y+a)=d(x,y). Thus open sets, closed sets, convergence and Cauchy sequences are inherited from the norm.
3. Convergence means norm difference tends to zero
A sequence x_n converges to x when
||x_n−x||→0.
The norm determines what “close” means. Different norms on the same underlying vector space can create different convergence behaviour in infinite dimensions.
4. Cauchy sequences express internal consistency before a limit is known
A sequence (x_n) is Cauchy if for every ε>0 there exists N such that m,n≥N implies
||x_n−x_m||<ε.
Every convergent sequence is Cauchy. The reverse implication is precisely the completeness question.
5. Banach spaces are complete normed spaces
A normed space X is a Banach space if every Cauchy sequence in X converges to a point of X.
Completeness prevents approximation procedures from “falling out” of the space. This is why completeness matters for iterative algorithms and existence proofs.
6. Rⁿ and Cⁿ are Banach under every norm
Every finite-dimensional normed vector space over R or C is complete.
Moreover, any two norms on a finite-dimensional vector space are equivalent: they generate the same convergent sequences and the same topology.
This statement fails dramatically in infinite-dimensional settings.
7. Norm equivalence means two-sided comparison
Norms ||·||_a and ||·||_b are equivalent if constants c,C>0 exist such that
c||x||_a≤||x||_b≤C||x||_a
for every x.
Equivalent norms have the same notions of convergence, Cauchy sequence, continuity and completeness.
8. ℓp spaces are fundamental sequence Banach spaces
For 1≤p<∞, define
ℓ^p={x=(x_n): Σ|x_n|^p<∞}
with norm ||x||_p=(Σ|x_n|^p)^{1/p}.
For p=∞, ℓ∞ consists of bounded sequences with norm ||x||∞=sup_n|x_n|.
Each ℓp space is Banach.
9. c₀ is a closed subspace of ℓ∞
Let c₀ be the sequences x_n→0, equipped with the sup norm.
If a sequence of c₀ elements converges uniformly in ℓ∞, its limit still tends to zero. Therefore c₀ is closed in ℓ∞.
A closed subspace of a Banach space is Banach, so c₀ is Banach.
10. Continuous-function spaces give the standard function example
On a compact interval [a,b], let C([a,b]) be the continuous scalar-valued functions with
||f||∞=max_{x∈[a,b]}|f(x)|.
A uniformly Cauchy sequence of continuous functions converges uniformly to a continuous function. Hence C([a,b]) is Banach in the sup norm.
11. A space can fail to be complete under an apparently natural norm
Consider C¹([0,1]) with only the sup norm ||f||∞.
There are sequences of C¹ functions converging uniformly to a continuous function that is not differentiable. The limit lies in C([0,1]) but not in C¹([0,1]).
Thus C¹ with the sup norm is not complete.
By contrast, the norm ||f||∞+||f’||∞ makes C¹([0,1]) Banach.
12. Lp spaces extend sequence norms to measurable functions
For 1≤p<∞, Lp consists of measurable functions modulo equality almost everywhere with
||f||_p=(∫|f|^p)^{1/p}<∞.
L∞ uses the essential supremum rather than the pointwise supremum.
Each Lp space is Banach. The “modulo almost everywhere” identification is essential because the norm cannot distinguish functions differing only on a null set.
13. A closed subspace of a Banach space is Banach
Let Y be a closed linear subspace of Banach space X. If (y_n) is Cauchy in Y, it is Cauchy in X and therefore converges to some x∈X.
Because Y is closed, x∈Y. Hence Y is complete.
Closedness is the key condition. Nonclosed subspaces can be incomplete even inside a complete ambient space.
14. Dense subspaces can be useful without being complete
Polynomials are dense in C([a,b]) under the sup norm by the Weierstrass approximation theorem, but the polynomial space itself is not complete in that norm.
A Cauchy sequence of polynomials can converge uniformly to a nonpolynomial continuous function.
Density supports approximation; completeness controls where limits live. They are different ideas.
15. Linear maps are continuous exactly when they are bounded
Let T:X→Y be linear between normed spaces. The following are equivalent:
- T is continuous everywhere;
- T is continuous at 0;
- there exists C≥0 with ||Tx||≤C||x|| for all x.
Such a map is called a bounded linear operator.
16. The operator norm is the smallest global Lipschitz constant
For bounded linear T:X→Y, define
||T||=sup_{||x||≤1}||Tx||=sup_{x≠0}||Tx||/||x||.
Then ||Tx||≤||T||||x|| for all x.
If Y is Banach, the space B(X,Y) of bounded linear maps is Banach in the operator norm.
17. Worked operator-norm example
On C([0,1]) with the sup norm, define
(Tf)(x)=∫₀^x f(t)dt.
Then |Tf(x)|≤x||f||∞≤||f||∞, so ||T||≤1.
Taking f≡1 gives ||Tf||∞=1, hence ||T||=1.
18. Not every linear map in infinite dimensions is continuous
In finite-dimensional normed spaces, every linear map is continuous.
In infinite dimensions, discontinuous linear maps exist if one allows arbitrary Hamel-basis constructions. They are usually nonconstructive and pathological, but their existence matters: linearity alone no longer guarantees analytic control.
19. The dual space collects bounded linear functionals
The continuous dual X* is the space of bounded linear maps φ:X→R or C.
It carries the norm
||φ||=sup_{||x||≤1}|φ(x)|.
X* is always Banach, even when X itself is not complete.
20. Coordinate functionals are bounded in finite dimensions
On Rⁿ with any norm, maps extracting a fixed coordinate are linear and continuous.
More generally, every linear functional on a finite-dimensional normed space is continuous.
Again, infinite dimensions require boundedness to be checked rather than assumed.
21. Hahn–Banach extends bounded linear functionals
A core Hahn–Banach consequence says that if Y is a subspace of normed space X and φ is a bounded linear functional on Y, then φ extends to a bounded linear functional on X without increasing its norm.
One useful corollary: for any nonzero x∈X there exists φ∈X* such that
||φ||=1 and φ(x)=||x||
in the real case, with the corresponding complex formulation.
Thus bounded functionals can separate points from zero and encode geometric information about the norm.
22. Baire category is the hidden engine behind three Banach-space theorems
The Baire Category Theorem says that a complete metric space cannot be written as a countable union of nowhere-dense closed sets.
This topological statement drives the Uniform Boundedness Principle, Open Mapping Theorem and Closed Graph Theorem.
Completeness therefore produces consequences that are far stronger than “Cauchy sequences converge”.
23. Uniform Boundedness converts pointwise bounds into an operator-norm bound
Let X be Banach and let {T_α}⊂B(X,Y). If for every fixed x∈X the set {||T_αx||} is bounded, then
sup_α ||T_α||<∞.
The completeness of the domain X is essential in the standard theorem.
24. The Open Mapping Theorem controls surjective bounded operators
If X and Y are Banach spaces and T:X→Y is bounded, linear and surjective, then T maps open sets to open sets.
A major corollary is the bounded inverse theorem: if T is a bounded linear bijection between Banach spaces, then T^{-1} is automatically bounded.
Algebraic invertibility plus Banach completeness gives analytic stability.
25. The Closed Graph Theorem turns geometric closure into continuity
For Banach spaces X,Y, a linear map T:X→Y is bounded iff its graph
G(T)={(x,Tx):x∈X}
is closed in X×Y, provided T is defined on all of X.
For unbounded operators with proper domains, closedness remains important but no longer means boundedness on the ambient space. R27.03 will separate those cases.
26. Quotient norms measure distance from a subspace
Let Y be a linear subspace of normed space X. On equivalence classes x+Y, define
||x+Y||=inf_{y∈Y}||x−y||.
This is a genuine norm exactly when Y is closed. If X is Banach and Y is closed, then X/Y is Banach.
The quotient norm turns approximation by Y into geometry: ||x+Y|| is the distance from x to Y.
27. Completion embeds an incomplete space into a Banach space
Every normed space X has a completion X̂: a Banach space containing an isometric dense copy of X.
The construction parallels completion of Q to R: represent new points by equivalence classes of Cauchy sequences.
Many function spaces are naturally obtained by completing a smoother or simpler dense class.
28. Completeness, compactness and finite dimension must not be conflated
In finite-dimensional normed spaces, closed and bounded sets are compact and the space is complete.
In infinite-dimensional normed spaces, the closed unit ball is generally not compact in the norm topology.
Banach means complete, not compact. Infinite-dimensional analysis requires new compactness ideas such as weak compactness and compact operators.
29. A useful noncompactness example
In ℓ², let e_n be the sequence with 1 in the nth coordinate and 0 elsewhere.
Each ||e_n||₂=1, but for m≠n,
||e_n−e_m||₂=√2.
Therefore the sequence has no norm-convergent subsequence. The unit ball is not norm compact.
30. Common theorem-boundary errors
- Complete does not mean every bounded sequence converges.
- Closed and bounded does not imply compact in general infinite-dimensional normed spaces.
- A dense subspace of a Banach space need not be complete.
- A linear map in infinite dimensions is not automatically continuous.
- Equivalent norms are automatic in finite dimensions, not in arbitrary infinite dimensions.
- C¹([0,1]) is not Banach under the sup norm alone.
- The Closed Graph Theorem for everywhere-defined maps must not be misapplied to unbounded operators with proper domains.
31. A dependable Banach-space workflow
Identify the vector space and scalar field. State the norm. Check that the intended limit process is measured in that norm rather than by informal pointwise intuition.
Before invoking a Banach theorem, verify completeness of every required space, closedness of relevant subspaces, and boundedness of each linear operator.
For an approximation problem, separate density from completeness. For inversion, separate algebraic bijectivity from bounded inverse control. For families of operators, distinguish pointwise boundedness from a uniform operator-norm estimate.
32. Independent practice: twenty questions
- State the three norm axioms.
- How does a norm induce a metric?
- Define norm convergence.
- Define a Cauchy sequence in a normed space.
- Define a Banach space.
- Why is every finite-dimensional normed space complete?
- What does equivalence of two norms mean?
- Define ℓ² and its norm.
- Define c₀ and explain why it is Banach in the sup norm.
- Why is C([0,1]) Banach in ||·||∞?
- Why is C¹([0,1]) not complete in the sup norm alone?
- State the boundedness criterion for a linear map T:X→Y.
- Define the operator norm.
- Compute the norm of Tf(x)=∫₀^x f(t)dt on C([0,1]) with sup norm.
- Define the continuous dual X*.
- State a Hahn–Banach extension consequence.
- State the Uniform Boundedness Principle in words.
- What does the Open Mapping Theorem imply for a bounded linear bijection between Banach spaces?
- State the Closed Graph Theorem for everywhere-defined maps.
- Why is the closed unit ball of ℓ² not norm compact?
33. Worked solutions and checks
1. Positive definiteness, absolute homogeneity and triangle inequality.
2. d(x,y)=||x−y||.
3. x_n→x when ||x_n−x||→0.
4. For every ε>0, eventually ||x_n−x_m||<ε for all sufficiently large m,n.
5. A complete normed vector space.
6. Finite-dimensional normed spaces are isomorphic to Rⁿ or Cⁿ and all norms are equivalent there; coordinate completeness transfers.
7. c||x||_a≤||x||_b≤C||x||_a for constants c,C>0.
8. ℓ²={x:Σ|x_n|²<∞}, with ||x||₂=(Σ|x_n|²)^{1/2}.
9. c₀ consists of sequences tending to 0; it is closed in Banach space ℓ∞.
10. A uniformly Cauchy sequence converges uniformly, and uniform limits of continuous functions are continuous.
11. Uniform limits of C¹ functions need not remain differentiable.
12. T is bounded iff some C satisfies ||Tx||≤C||x|| for all x; for linear maps this is equivalent to continuity.
13. ||T||=sup_{||x||≤1}||Tx||.
14. ||T||=1.
15. The Banach space of bounded linear scalar-valued functionals on X.
16. A bounded functional on a subspace extends to the whole normed space with the same norm.
17. A pointwise-bounded family of bounded linear operators on a Banach domain is uniformly bounded in operator norm.
18. The inverse is automatically bounded.
19. An everywhere-defined linear map between Banach spaces is bounded iff its graph is closed.
20. The unit vectors e_n all have norm 1 but pairwise distance √2, so no subsequence can be Cauchy and hence none converges in norm.
34. Where this guide hands off
R27.02 adds inner-product geometry. In Hilbert spaces, closed subspaces admit orthogonal projections, bounded linear functionals have vector representatives, orthonormal systems support Fourier expansion, and least-squares approximation becomes a geometric theorem rather than a matrix trick.
Sources and further study
For a university functional-analysis route, see MIT OpenCourseWare 18.102 Introduction to Functional Analysis. For metric completeness and compactness prerequisites, use BTT’s Metric Spaces and Examples and Completeness of the Real Numbers.