query: Existence of complement of a subspace without Zorn's lemma
positive_passages.docid: 106670
positive_passages.text: Existence of vector space complement and axiom of choice
negative_passages.docid: 123613
negative_passages.text: Double Complement of a set proof
negative_passages.docid: 12212
negative_passages.text: Showing a certain subspace of Hilbert space is dense
negative_passages.docid: 15560
negative_passages.text: Prove that the union of three subspaces of $V$ is a subspace iff one of the subspaces contains the other two.
negative_passages.docid: 144572
negative_passages.text: Proof that ideals in $C[0,1]$ are of the form $M_c$ that should not involve Zorn's Lemma
negative_passages.docid: 67191
negative_passages.text: When is the union of a family of subspaces of a vector space also a subspace?
negative_passages.docid: 67259
negative_passages.text: Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?
negative_passages.docid: 15576
negative_passages.text: Addition of two subspaces
negative_passages.docid: 146664
negative_passages.text: The union of two linearly independent subsets
negative_passages.docid: 96717
negative_passages.text: Hahn-Banach Theorem for separable spaces without Zorn's Lemma
negative_passages.docid: 122463
negative_passages.text: How to show the that a set $A$ nowhere dense is equivalent to the complement of $A$ containing a dense open set?
negative_passages.docid: 76148
negative_passages.text: Show that $\exists c \in \mathbb{R} : f(c) = 0$
negative_passages.docid: 33642
negative_passages.text: Finite dimensional subspace of $C([0,1])$
negative_passages.docid: 73154
negative_passages.text: Compact subset in colimit of spaces
negative_passages.docid: 150605
negative_passages.text: Existence of a regular uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$
negative_passages.docid: 14550
negative_passages.text: Every Hilbert space has an orthonomal basis - using Zorn's Lemma
negative_passages.docid: 22295
negative_passages.text: Subspace of a separable space is separable