query: $\exists S \subset \mathbb{R}$ s.t. every $x \in \mathbb{R}$ can be represented as a unique linear combination of numbers with rational coefficients
positive_passages.docid: 154724
positive_passages.text: Every vector space has a basis
negative_passages.docid: 63538
negative_passages.text: Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$
negative_passages.docid: 56985
negative_passages.text: Show that every positive integer has a unique representation in the form $50k+l$...?
negative_passages.docid: 76148
negative_passages.text: Show that $\exists c \in \mathbb{R} : f(c) = 0$
negative_passages.docid: 63313
negative_passages.text: Uniqueness of Powerseries in an arbitrarily small neighborhood of $0$ in $\mathbb C$
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: 19954
negative_passages.text: Perfect set without rational numbers
negative_passages.docid: 12701
negative_passages.text: Perfect set without rationals
negative_passages.docid: 2287
negative_passages.text: If a set is linearly independent, a finite subset of that set is also linearly independent
negative_passages.docid: 150674
negative_passages.text: Uniqueness in existence of a bilinear form
negative_passages.docid: 723
negative_passages.text: Is there a constructive way to exhibit a basis for $\mathbb{R}^\mathbb{N}$?
negative_passages.docid: 157924
negative_passages.text: $\forall x\in\mathbb{N}[(\exists y\in\mathbb{N}(2\leq y\land y<x\land y\ |\ x))\to(\exists z\in\mathbb{N}(2\leq z\land z\leq\sqrt x\land z\ |\ x))]$
negative_passages.docid: 69499
negative_passages.text: Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0\alpha$.
negative_passages.docid: 145471
negative_passages.text: $\Gamma \subset \mathbb{R}^{+}$ is uncountable. Can we choose a sequence from $\Gamma$ of which the sum is $\infty$
negative_passages.docid: 60655
negative_passages.text: $\ker T\subset \ker S\Rightarrow S=rT$ when $S$ and $T$ are linear functionals
negative_passages.docid: 99854
negative_passages.text: Must an uncountable subset of R have uncountably many accumulation points?
negative_passages.docid: 73927
negative_passages.text: Is $\mathbb{R}$ a subset of $\mathbb{R}^2$?