[#P2926] An infinite-dimensional Banach space where every operator attains its norm
Problem. Does there exist an infinite-dimensional real or complex Banach space \(X\) such that every bounded linear operator \(T:X\to X\) attains its operator norm?
1Context
Approximation-property obstructions, James-boundary witnesses, and explicit nonattaining operators for candidate spaces can be reused. Known necessary conditions narrow any positive example to an unusual separable reflexive space.
2Problem setup
Definition 1 (A bounded operator T attains its norm if there). A bounded operator T attains its norm if there is x in X with ||x||=1 and ||Tx||=||T||, where ||T||=sup_{||y||=1}||Ty||.
Definition 2 (The scalar field). The scalar field is R or C with its usual absolute value; Banach spaces over finite or other valued fields are outside the question.
Remark 1. Approximation-property obstructions, James-boundary witnesses, and explicit nonattaining operators for candidate spaces can be reused. Known necessary conditions narrow any positive example to an unusual separable reflexive space.
3What counts as a solution
- Construct an infinite-dimensional real or complex Banach space X and prove that every T in L(X) attains its norm, or prove that every infinite-dimensional real or complex Banach space admits a bounded self-operator that fails to attain its norm.
- A positive construction must cover all bounded self-operators. Verifying compact, finite-rank, or scalar-plus-compact subclasses alone does not settle the target.
1Status
Current status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. A 2024 specialist lecture still lists Ostrovskii's 2005 question as an open problem. Dantas, Jung, and Martinez-Cervantes prove new sufficient conditions for non-norm-attaining operators in 2021, without resolving the remaining separable reflexive spaces lacking the relevant approximation properties. Construct an infinite-dimensional real or complex Banach space X and prove that every T in L(X) attains its norm, or prove that every infinite-dimensional real or complex Banach space admits a bounded self-operator that fails to attain its norm.[1]
1Records
Notes and companion material
Original intake status. UNKNOWN as of 2026-07-31. A 2024 specialist lecture still lists Ostrovskii's 2005 question as an open problem. Dantas, Jung, and Martinez-Cervantes prove new sufficient conditions for non-norm-attaining operators in 2021, without resolving the remaining separable reflexive spaces lacking the relevant approximation properties.
- The 2021 primary paper arXiv:2102.06452 gives new criteria producing non-norm-attaining operators and extends the Holub-Mujica obstruction under bounded compact approximation assumptions. It does not rule out every infinite-dimensional X.
- A 2024 ICMAT lecture, Questions and results around James' Theorem, displays the exact statement as Problem (M. I. Ostrovskii, 2005) and summarizes the remaining necessary conditions, providing dated current-status evidence.
- A 2026-07-27 exact-statement and citation search found no later construction or impossibility theorem. A local corpus search for all operators norm-attaining and L(X)=NA(X) found no duplicate.
Recorded example 1. On ell_2, the diagonal operator T(e_n)=(1-1/n)e_n has norm one and attains no unit vector at that norm, so ell_2 does not have the requested property.
Recorded example 2. If X is nonreflexive, James' theorem gives a non-norm-attaining functional; multiplying it by a fixed nonzero vector produces a rank-one self-operator that fails to attain its norm.
How the 2 records connect
ProblemAn infinite-dimensional Banach space where every operator attains its norm
2See also
- Openness of convolution on l1 of the integersfunctional analysis
- Logarithmic dimension for almost-equilateral sets in Banach spacesbanach spaces
- Planar drums whose spectra differ only finitelymathoverflow source
How to cite
TheoremDB contributors, “An infinite-dimensional Banach space where every operator attains its norm,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/all-operators-norm-attaining-banach-spaceThis page as plain text: all-operators-norm-attaining-banach-space.md
This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net[1], current as of July 31, 2026.
1References
- Packet source. MathOverflow: Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?. Question 232291 and all thirteen visible comments were checked on 2026-07-27. The live unclosed page has no answers or accepted answer and reproduces Ostrovskii's 2005 problem. Question 232291 and all thirteen visible comments were checked on 2026-07-27. The live unclosed page has no answers or accepted answer and reproduces Ostrovskii's 2005 problem. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. A 2024 specialist lecture still lists Ostrovskii's 2005 question as an open problem. Dantas, Jung, and Martinez-Cervantes prove new sufficient conditions for non-norm-attaining operators in 2021, without resolving the remaining separable reflexive spaces lacking the relevant approximation properties.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For An infinite-dimensional Banach space where every operator attains its norm: UNKNOWN as of 2026-07-27. A 2024 specialist lecture still lists Ostrovskii's 2005 question as an open problem. Dantas, Jung, and Martinez-Cervantes prove new sufficient conditions for non-norm-attaining operators in 2021, without resolving the remaining separable reflexive spaces lacking the relevant approximation properties.Source named by the research packet.
- MathOverflow: Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?, source checked for the TheoremDB status review (2026-07-31). Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:2102.06452, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. A 2024 specialist lecture still lists Ostrovskii's 2005 question as an open problem. Dantas, Jung, and Martinez-Cervantes prove new sufficient conditions for non-norm-attaining operators in 2021, without resolving the remaining separable reflexive spaces lacking the relevant approximation properties.Also cited at Full preprint relevant to An infinite-dimensional Banach space where every operator attains its norm.Source used to assess the problem's recorded status.For An infinite-dimensional Banach space where every operator attains its norm: UNKNOWN as of 2026-07-27. A 2024 specialist lecture still lists Ostrovskii's 2005 question as an open problem. Dantas, Jung, and Martinez-Cervantes prove new sufficient conditions for non-norm-attaining operators in 2021, without resolving the remaining separable reflexive spaces lacking the relevant approximation properties.
- MathOverflow: Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?, source checked for the TheoremDB status review (2026-07-31). Status evidence identified in the source record and checked at the linked publication. ↗website · primary source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. A 2024 specialist lecture still lists Ostrovskii's 2005 question as an open problem. Dantas, Jung, and Martinez-Cervantes prove new sufficient conditions for non-norm-attaining operators in 2021, without resolving the remaining separable reflexive spaces lacking the relevant approximation properties.Also cited at Part II, operator versions of James’ theorem, May 20, 2024 lecture slides.Source used to assess the problem's recorded status.For An infinite-dimensional Banach space where every operator attains its norm, this source surveys the current operator-level question and recent partial results without claiming a construction.
- V. K. Maslyuchenko and A. M. Plichko, editors, “Some Open Problems on Functional Analysis and Function Theory,” Extracta Mathematicae 20(1) (2005), 51-70; M. I. Ostrovskii, Problem 12.1, pages 65-66. Status evidence identified in the source record and checked at the linked publication. ↗website · primary source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. A 2024 specialist lecture still lists Ostrovskii's 2005 question as an open problem. Dantas, Jung, and Martinez-Cervantes prove new sufficient conditions for non-norm-attaining operators in 2021, without resolving the remaining separable reflexive spaces lacking the relevant approximation properties.Also cited at Problem 12.1, ‘Norm-attaining operators,’ pages 65-66.Source used to assess the problem's recorded status.For An infinite-dimensional Banach space where every operator attains its norm, this source states Ostrovskii’s exact infinite-dimensional norm-attainment problem and its 2005 necessary conditions.
An original CC0 textbook restatement motivated by the cited MathOverflow question; the scalar-field and self-operator conventions are explicit.