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MathidiaMathematics Academy
Functional methods
Lesson 22 of 75
Easy practice

Complete normed spaces

Objective. Use completeness in functional settings.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
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Learn: the key idea

A Banach space is a normed vector space that is complete, meaning every Cauchy sequence converges to a limit within the space itself, with no 'gaps' like the rationals have inside the reals. Completeness is essential for many functional analysis tools, such as the contraction mapping theorem, which guarantees unique fixed points for contractions on complete spaces. The space C[0,1] of continuous functions with the sup norm is a Banach space because uniform limits of continuous functions are continuous, while the space of polynomials with the sup norm is not complete, since polynomial approximations can converge uniformly to non-polynomial functions like eˣ.

Worked example

Explain why the space ℚ of rational numbers with the usual absolute value norm fails to be a complete normed space.

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Illustrated card showing ?
  1. 1Completeness requires every Cauchy sequence in ℚ to converge to a limit in ℚ
  2. 2Consider the sequence of rational truncations of √2, e.g. 1, 1.4, 1.41, 1.414,...
  3. 3This sequence is Cauchy in ℚ but its limit √2 is irrational
  4. 4Since the limit lies outside ℚ, ℚ is not complete

Answer: ℚ is not complete because Cauchy sequences can converge to irrational limits

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Common mistake. Students assume every normed space is automatically complete, but the space of polynomials on [0,1] with the sup norm is an incomplete normed space, since the Cauchy sequence of partial sums of the Taylor series for eˣ converges to eˣ, which lies outside the space of polynomials.

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