query: Bolzano-Weistrass theorem true in any vector space?
positive_passages.docid: 148999
positive_passages.text: In what spaces does the Bolzano-Weierstrass theorem hold?
negative_passages.docid: 151425
negative_passages.text: Does $\mathbb{R}^n$ have a real vector space structure with dimension other than $n$?
negative_passages.docid: 77036
negative_passages.text: Normed vector spaces over finite fields
negative_passages.docid: 122121
negative_passages.text: Is there a vector space that cannot be an inner product space?
negative_passages.docid: 31069
negative_passages.text: Smallest and largest values of $\|\vec{v}-\vec{w}\|$
negative_passages.docid: 60685
negative_passages.text: The vector space formed on C[0,1] and the norm $(\int_{0}^{1}|f(t)|^2 dt)^{1/2} $
negative_passages.docid: 53850
negative_passages.text: Proof that every normed vector space is a topological vector space
negative_passages.docid: 109200
negative_passages.text: Baby Rudin theorem 2.41
negative_passages.docid: 123505
negative_passages.text: how to prove vector norm equivalence in finite dimensional space($\mathbb{R}^{n}$)?
negative_passages.docid: 99138
negative_passages.text: Question about Grassmannian, most vectors in $\bigwedge^k V$ are not completely decomposable?
negative_passages.docid: 27478
negative_passages.text: Proof that two basis of a vector space have the same cardinality in the infinite-dimensional case
negative_passages.docid: 58447
negative_passages.text: Is axiom of choice required for there to be an infinite linearly independent set in a (non-finite-dimensional) vector space?
negative_passages.docid: 143517
negative_passages.text: Isomorphisms Between a Finite-Dimensional Vector Space and its Dual
negative_passages.docid: 34865
negative_passages.text: prove Eckart-Young-Minsky theorem for Frobinius norm.
negative_passages.docid: 117986
negative_passages.text: Are there linear transformations from vector spaces over different fields?
negative_passages.docid: 34206
negative_passages.text: Quadratic form under $\mathbb{C}$ finite-dimension vector space always have a nonzero vector that yields zero?
negative_passages.docid: 24478
negative_passages.text: What is the name of this theorem, and are there any caveats?