query: Prove that a subpace of all functions from a set $S$ to a field $F$ has a basis $f_1,...,f_n$ which is dual with some elements $x_1,...,x_n$.
positive_passages.docid: 134065
positive_passages.text: A question about an $n$-dimensional subspace of $\mathbb{F}^{S}$.
negative_passages.docid: 23125
negative_passages.text: Prove that there is a bijection between the set of all subsets of $X$, $P(X)$, and the set of functions from $X$ to $\{0,1\}$.
negative_passages.docid: 116768
negative_passages.text: Problem 5, sec. 13 in Munkres' Topology, 2nd ed.: How to prove the assertion if $\mathscr{A}$ is a subbasis?
negative_passages.docid: 144250
negative_passages.text: subideals of an ideal
negative_passages.docid: 16512
negative_passages.text: Product of $T_1$ spaces is $T_1$
negative_passages.docid: 89271
negative_passages.text: If $F$ is a field, then $F[x]$ is a principal ideal domain proof question
negative_passages.docid: 154724
negative_passages.text: Every vector space has a basis
negative_passages.docid: 21366
negative_passages.text: Help on the relationship of a basis and a dual basis
negative_passages.docid: 88278
negative_passages.text: Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
negative_passages.docid: 33563
negative_passages.text: Is the axiom of choice used in the proof that every open set is the union of basis elements (Mukres, Lemma 13.1)
negative_passages.docid: 107383
negative_passages.text: Derived sets - prove $(A \cup B)' = A' \cup B'$
negative_passages.docid: 145140
negative_passages.text: Subfield of $\mathbb{Q}(\sqrt[n]{a})$
negative_passages.docid: 151295
negative_passages.text: Proving $|S \times \mathbb{N}|=|S|$ when $S$ is an infinite set
negative_passages.docid: 98577
negative_passages.text: Show that the set $\mathbb{Q}(\sqrt{p})=\{a+b\sqrt{p}; a,b,p\in\mathbb{Q},\sqrt{p}\notin \mathbb{Q}\}$ is a field
negative_passages.docid: 98256
negative_passages.text: Subfields of $\mathbb{Q}(\sqrt{2},\sqrt{3})$
negative_passages.docid: 106284
negative_passages.text: Establishing a Basis for $F[x]/(f(x))$
negative_passages.docid: 125962
negative_passages.text: The set $A'$ is closed in a $T_1$ space