Q57·Section II (Mathematics)·57 of 190Finitely-supported subspaces of ℓ2\ell^2ℓ2 never algebraically sum to all of ℓ2\ell^2ℓ2Let Vk={(an)∈ℓ2:an=0 for all n>k}V_k=\{(a_n)\in\ell^2:a_n=0\text{ for all }n>k\}Vk={(an)∈ℓ2:an=0 for all n>k}. Then: OptionsAVkV_kVk is finite dimensional for all kkk and ℓ2\ell^2ℓ2 is a direct sum of {Vk∣k∈N}\{V_k\mid k\in\mathbb{N}\}{Vk∣k∈N}BVkV_kVk is finite dimensional for all kkk and ℓ2\ell^2ℓ2 is a sum of {Vk∣k∈N}\{V_k\mid k\in\mathbb{N}\}{Vk∣k∈N} and is not a direct sumCVkV_kVk is infinite dimensional for all kkk and ℓ2\ell^2ℓ2 is a sum of {Vk∣k∈N}\{V_k\mid k\in\mathbb{N}\}{Vk∣k∈N}DVkV_kVk is finite dimensional for all kkk and ℓ2\ell^2ℓ2 is not a sum of {Vk∣k∈N}\{V_k\mid k\in\mathbb{N}\}{Vk∣k∈N}Pick an option to check your answer.Show answerShow solution Report a problem with this question← Q56All questionsQ58 →Keyboard: A–D answer · S solution · ←/→ previous/next