Questions: Completeness of L^p Spaces (Riesz-Fischer Theorem)

Question 1

Why must L^p functions be defined as equivalence classes (identifying functions that agree almost everywhere), rather than as individual measurable functions?

• A. It is a notational convenience that could be dropped without affecting the theory
• B. Without this identification, ‖f‖_p = 0 would not imply f = 0, so the 'norm' would only be a seminorm, not a genuine norm
• C. The identification is required only for p = 2 (the Hilbert space case), not for general p
• D. It ensures every L^p function has a pointwise-defined representative everywhere on the domain

Answer: (B)

A genuine norm requires ‖f‖_p = 0 ⟹ f = 0 (positive-definiteness). But ‖f‖_p = 0 in the integral sense only forces f = 0 almost everywhere — functions like the indicator of the rationals on [0,1] have L^p norm zero but are not the zero function. Quotienting by null sets promotes the seminorm to a true norm and is structurally necessary, not pedantic. Without it, the space is not a normed space and the question of completeness is not even well-posed.

Question 2

The Riesz-Fischer proof extracts a rapidly converging subsequence with ‖f_{n_{k+1}} − f_{n_k}‖_p ≤ 2^{−k} rather than working directly with the original Cauchy sequence. What is the purpose of this extraction?

• A. The original Cauchy sequence always diverges, so a subsequence must be found to locate any limit
• B. To avoid applying Minkowski's inequality, which is only valid for finite sums
• C. The summability of 2^{−k} allows the partial sums of incremental differences to be controlled, enabling the monotone convergence theorem to show the constructed series converges a.e. and in L^p
• D. Cauchy sequences in L^p have no pointwise values, so a pointwise-convergent subsequence must be constructed separately

Answer: (C)

A Cauchy sequence only guarantees that gaps between terms become small — not that they are summable. By extracting a subsequence where ‖f_{n_{k+1}} − f_{n_k}‖_p ≤ 2^{−k}, the total variation ∑2^{−k} = 1 is finite. Minkowski's inequality then controls the partial sums, and the monotone convergence theorem shows the telescoping series converges both a.e. and in L^p norm. Without this extraction, the a.e. convergence argument does not go through and you cannot construct the candidate limit function.

Question 3

Nearly every normed vector space is complete, so proving the Minkowski inequality for L^p is sufficient to establish that L^p is a Banach space.

• T. True
• F. False

Answer: False

False — being a normed vector space does not imply completeness. The rationals ℚ with the absolute value form a normed vector space, yet ℚ is not complete: a sequence of rational approximations to √2 is Cauchy but has no limit in ℚ. Completeness is an additional property that must be proved separately. The Minkowski inequality establishes L^p is a normed space; the Riesz-Fischer theorem is the separate work required to prove completeness.

Question 4

The Riesz-Fischer theorem guarantees that every Cauchy sequence in L^p converges to an L^p function in the L^p norm, but it does not guarantee pointwise convergence everywhere on the domain.

• T. True
• F. False

Answer: True

Correct. L^p convergence (‖f_n − f‖_p → 0) is a statement about integrated differences, not pointwise behavior. A Cauchy sequence in L^p can fail to converge pointwise at every individual point. The Riesz-Fischer proof does produce a subsequence that converges almost everywhere, but 'almost everywhere' excludes a null set that may still be dense (like the rationals in [0,1]). The theorem guarantees norm convergence of the full sequence, and a.e. convergence for a subsequence — not pointwise convergence everywhere.

Question 5

Explain why the completeness of L^p matters for harmonic analysis, giving a concrete example of a result that depends on it.

Model answer: Completeness guarantees that limit processes stay within L^p. In harmonic analysis, the key example is L^2: the partial sums of the Fourier series of an L^2 function form a Cauchy sequence in L^2, and completeness is exactly what allows us to conclude this sequence converges to a function that is itself in L^2. Without completeness, the limit might 'fall out' of the space and the convergence theorem would have no meaning. More broadly, spectral theory, variational methods, and fixed-point arguments all require that Cauchy sequences built during the proof converge to something still inside the space.

Completeness is the property that makes analysis possible within a function space: you can take limits and trust the result is still the kind of object you are working with. The Fourier series example is clean — partial sums are in L^2, the sequence is Cauchy in L^2 norm, and Riesz-Fischer is precisely what lets you conclude the limit is in L^2. Without it, every convergence argument inside L^p would require a separate verification that the limit belongs to the space.