Changed norms, compact families and continuous measurements
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 180 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Separate a fixed finite coordinate range from the whole infinite sequence.
Justify membership before using a norm or completeness theorem.
State uniform tail conditions with their quantifiers; pointwise tail statements are weaker.
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Let X be the real sequences x for which Σ from k=1 to infinity of 2⁻ᵏ|xₖ| is finite, with that sum as norm. Prove X is complete by a concrete isometry with ℓ1. Prove finitely supported sequences are dense in X. Regard ordinary ℓ1 as a subspace of X with this new weighted norm: is it closed or complete? Use truncations of the constant-one sequence to prove your conclusions; do not confuse this norm with the usual ℓ1 norm.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Fix 1≤p<∞. Prove that a subset A of ℓp is totally bounded if and only if it is bounded and for every ε>0 there is K with ‖x−P_Kx‖p<ε for all x∈A. You may use that a complete totally bounded metric space is compact, but justify the finite head approximation rather than assume infinite-dimensional bounded sets are compact. Deduce a compactness criterion for closed A. Apply it to {eₘ:m≥1} and {0}∪{eₘ/m:m≥1}.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
On finitely supported real sequences define S(x)=Σxₖ and T(x)=Σ2⁻ᵏxₖ. Equip this domain with the ℓ2 norm. Show that S has no continuous linear extension to ℓ2. Prove T does have a unique continuous linear extension, give its infinite-series formula, and prove its exact operator norm is 1/√3. Explain why density alone did not suffice for S.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Let X be the real sequences x for which Σ from k=1 to infinity of 2⁻ᵏ|xₖ| is finite, with that sum as norm. Prove X is complete by a concrete isometry with ℓ1. Prove finitely supported sequences are dense in X. Regard ordinary ℓ1 as a subspace of X with this new weighted norm: is it closed or complete? Use truncations of the constant-one sequence to prove your conclusions; do not confuse this norm with the usual ℓ1 norm.
Fix 1≤p<∞. Prove that a subset A of ℓp is totally bounded if and only if it is bounded and for every ε>0 there is K with ‖x−P_Kx‖p<ε for all x∈A. You may use that a complete totally bounded metric space is compact, but justify the finite head approximation rather than assume infinite-dimensional bounded sets are compact. Deduce a compactness criterion for closed A. Apply it to {eₘ:m≥1} and {0}∪{eₘ/m:m≥1}.
On finitely supported real sequences define S(x)=Σxₖ and T(x)=Σ2⁻ᵏxₖ. Equip this domain with the ℓ2 norm. Show that S has no continuous linear extension to ℓ2. Prove T does have a unique continuous linear extension, give its infinite-series formula, and prove its exact operator norm is 1/√3. Explain why density alone did not suffice for S.
