query: Show that if $M$ is closed ,then $(M^\perp)^\perp=M$
positive_passages.docid: 19424
positive_passages.text: Showing $(M^\perp)^\perp=\overline{M}$
negative_passages.docid: 80892
negative_passages.text: Does $\wp(A \cap B) = \wp(A) \cap \wp(B)$ hold? How to prove it?
negative_passages.docid: 125868
negative_passages.text: Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
negative_passages.docid: 78471
negative_passages.text: Closure of $\{\frac{1}{n}\}:=M$
negative_passages.docid: 120010
negative_passages.text: Show that $\aleph_{\alpha}^{\aleph_{1}} = \aleph_{\alpha}^{\aleph_{0}}\cdot 2^{\aleph_1}$
negative_passages.docid: 143457
negative_passages.text: Is product of two closed sets closed?
negative_passages.docid: 3109
negative_passages.text: Prove that $(p+q)^m \leq p^m+q^m$
negative_passages.docid: 74792
negative_passages.text: $(X,\tau)$ needs to be $T_1$ in order to guarantee that $A'$ is closed?
negative_passages.docid: 146093
negative_passages.text: Why if $n \mid m$, then $(a^n-1) \mid (a^m-1)$?
negative_passages.docid: 141762
negative_passages.text: The direct sum of two closed subspace is closed? (Hilbert space)
negative_passages.docid: 29231
negative_passages.text: Proof of : $x^{n}\equiv x^{\varphi(m)+[n \bmod \varphi(m)]} \mod m$
negative_passages.docid: 8414
negative_passages.text: Proof verification: Set of partial limits of a sequence is closed
negative_passages.docid: 124685
negative_passages.text: How to show $(mn)!$ is divisible by $(m!)^n$?
negative_passages.docid: 156671
negative_passages.text: Necessary and sufficient condition for $x^n - y^m$ to be irreducible in $\Bbb C[x,y]$
negative_passages.docid: 145423
negative_passages.text: prove that if $1 \le m \le n$ then $\left(1+\frac1n\right)^m < 1+\frac mn + \left(\frac mn\right)^2$
negative_passages.docid: 47680
negative_passages.text: Prove that the union of the interior of a set and the boundary of the set is the closure of the set
negative_passages.docid: 65162
negative_passages.text: Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise.