Normed Spaces, Completeness and Banach Spaces
A vector space with a length function, plus the question of whether every Cauchy sequence in it converges. Completeness is what lets an existence proof finish by taking a limit and knowing the limit is still in the space.
What a learner can do afterwards
- Verify the norm axioms for a proposed norm
- Give a normed space that is not complete and name the limit it is missing
- Explain why all norms agree in finite dimensions and why that argument fails in infinite dimensions
1 · Read
You start with a length function called a norm, which follows three rules: only the zero vector has length zero, scaling a vector scales its length by the absolute value, and the length of a sum never exceeds the sum of the lengths. You verify each rule directly on examples like the sum of absolute values or the square root of the sum of squares. The third rule is the triangle inequality.
You can watch completeness fail in the polynomials on the interval from 0 to 1 with the sup norm, which records the largest absolute value. The Taylor partial sums of the exponential form a Cauchy sequence there, and their uniform limit is the exponential itself. Since the exponential is not a polynomial, the space is missing the very limit it needs.
In finite dimensions you get norm equivalence for free: the unit sphere is compact, so any other norm attains a positive minimum and maximum on it, and those extremes give the comparison bounds. In infinite dimensions the closed unit ball need not be compact, so that argument collapses. On polynomials the sup norm and the L1 norm drift apart with no fixed bounds to compare them.
Completeness is what lets your existence proof finish: you build a Cauchy sequence of approximations, take its limit, and know the limit still lies in your space. A complete normed space earns the name Banach space. You will often use the fact that a closed subspace of a Banach space is again Banach.
A norm supplies lengths, completeness keeps every limit inside the space, and Banach spaces are where approximation arguments land safely.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.