query: If $\|u+\lambda v\| \ge \|u\| \forall \lambda$ then $u \perp v$
positive_passages.docid: 34147
positive_passages.text: $\|u\|\leq \|u+av\| \Longrightarrow \langle u,v\rangle=0$
negative_passages.docid: 35564
negative_passages.text: Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$
negative_passages.docid: 135694
negative_passages.text: Proving $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$
negative_passages.docid: 91087
negative_passages.text: For which values $r, s \in \mathbb R$ is the set $\{u, v, w\}$ linearly independent?
negative_passages.docid: 27564
negative_passages.text: Prove that $\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\|V-I\|$
negative_passages.docid: 102959
negative_passages.text: Show that $f$ is convex if and only if $f\left( \sum_{i=1}^m\lambda_ix_i \right) \leq \sum_{i=1}^m\lambda_if(x_i)$
negative_passages.docid: 48291
negative_passages.text: Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $
negative_passages.docid: 131609
negative_passages.text: Prove that $Au\cdot v = u\cdot A^Tv$
negative_passages.docid: 95379
negative_passages.text: Prove that $V = V_1 \oplus V_2$ in the following
negative_passages.docid: 74623
negative_passages.text: Associative rule of subspaces and vector spaces: $(U+V) \cap W = U + (V \cap W)$
negative_passages.docid: 49073
negative_passages.text: how to obtain solution for $u_t+u=u_{xx}$
negative_passages.docid: 10859
negative_passages.text: Prove $\sup(f+g) \le \sup f + \sup g$
negative_passages.docid: 123483
negative_passages.text: Prove that $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A + B)$
negative_passages.docid: 20789
negative_passages.text: If $\Sigma \vdash \varphi$ then $\Sigma \vdash \forall x \varphi$. Why when $(\varphi \implies \forall x \varphi)$ is not true?
negative_passages.docid: 102391
negative_passages.text: Abel summability under the hypothesis $a_n \ge 0$
negative_passages.docid: 83279
negative_passages.text: How can I find such a $z$? (Injectivity of $T(z) = \lambda z + \mu\bar{z}$)
negative_passages.docid: 158551
negative_passages.text: If $X \perp Y$ and $X+Y \in L^1$ then $X,Y \in L^1$