Questions: Banach Spaces

Question 1

The rational numbers ℚ with the absolute value metric form a normed vector space but NOT a Banach space. What is the precise reason?

• A. ℚ is not a vector space because it lacks additive inverses for all elements
• B. The absolute value does not satisfy the triangle inequality on ℚ
• C. Cauchy sequences of rational numbers can converge to irrational limits that lie outside ℚ
• D. ℚ is not closed under scalar multiplication by real numbers

Answer: (C)

ℚ is a perfectly valid normed vector space (over itself). It fails the Banach condition because it is not complete: there exist Cauchy sequences in ℚ whose limit is irrational (e.g., the decimal approximations to √2). A Cauchy sequence is 'trying to converge' — its terms get arbitrarily close — but in ℚ, the limit may not exist within the space. Banach spaces require that every Cauchy sequence converge to a limit that is *also* in the space. This is exactly the gap between ℚ and ℝ.

Question 2

Why does completeness matter specifically when the normed space consists of *functions* (like C[a,b] or Lᵖ) rather than finite-dimensional vectors?

• A. Completeness ensures the triangle inequality holds for function norms, which it does not in incomplete spaces
• B. Without completeness, Cauchy sequences of well-behaved functions can converge to something discontinuous or outside the space, breaking analytic machinery
• C. Completeness prevents function sequences from diverging to infinity, which vectors cannot do
• D. In finite dimensions all normed spaces are automatically complete, so completeness only needs to be checked for function spaces

Answer: (B)

In function spaces, a Cauchy sequence in the norm is a sequence of functions whose pairwise distance (measured by the norm) goes to zero — they are 'trying to converge' to some limiting function. Without completeness, that limit may fail to be continuous, integrable, or in the space at all. For example, a sequence of continuous functions converging pointwise to a discontinuous function is Cauchy in many norms; completeness (under the sup norm for C[a,b], or Lᵖ norm for Lᵖ) guarantees the limit lands where it should. Option D is also true for finite dimensions, but that is not why completeness matters in infinite dimensions.

Question 3

Any normed vector space in which most convergent sequence is Cauchy is automatically a Banach space.

• T. True
• F. False

Answer: False

In any metric space, every convergent sequence is Cauchy — this is trivially true and requires no completeness assumption. The Banach condition is the non-trivial *converse*: every Cauchy sequence must *converge* (to a point within the space). Completeness is a property of the space, not just a property of convergent sequences. An incomplete normed space trivially satisfies 'convergent implies Cauchy' while still lacking completeness.

Question 4

The space C[a,b] of continuous functions on a closed interval, equipped with the supremum norm ‖f‖ = sup|f(x)|, is a Banach space.

• T. True
• F. False

Answer: True

The key fact is that a uniformly convergent (in sup norm) sequence of continuous functions converges to a continuous function — so the limit stays in C[a,b]. This is the essential property: the space is closed under limits of Cauchy sequences. Contrast this with the space of polynomials on [a,b] under the sup norm: a uniformly convergent sequence of polynomials can converge to a non-polynomial (like e^x), so that space is not complete.

Question 5

Why do the major theorems of functional analysis — the Banach contraction mapping theorem, the open mapping theorem, the Hahn-Banach theorem — require the spaces involved to be Banach spaces rather than arbitrary normed spaces?

Model answer: These theorems depend on the ability to take limits and guarantee they land within the space. The contraction mapping theorem constructs a fixed point as the limit of an iterated sequence x_{n+1} = T(x_n); if the space is incomplete, the sequence may be Cauchy but converge to something outside the space, so no fixed point exists within it. The open mapping theorem's proof constructs a convergent series whose sum must lie in the space. Without completeness, the analytic machinery of taking limits, summing series, and applying iterative procedures fails to produce results that stay in the space you started with.

Banach spaces play the same role in infinite-dimensional analysis that ℝ plays in classical analysis: they are the complete arenas where limits work. Just as calculus requires ℝ rather than ℚ (so that Cauchy sequences converge), functional analysis requires Banach spaces so that fixed-point iterations, linear approximations, and operator limits converge to genuine elements of the space.