Let U β R n U\subset \mathbb{R}^n U β R n H β 1 ( U ) H^{-1}(U) H β 1 ( U ) H 0 1 ( U ) H_0^1(U) H 0 1 β ( U ) β₯ F β₯ H β 1 ( U ) : = sup β‘ { β£ β¨ F , v β© β£ : v β H 0 1 ( U ) , β₯ v β₯ H 0 1 ( U ) β€ 1 } \|F\|_{H^{-1}(U)}:=\sup\{ |\langle F,v\rangle| : v\in H_0^1(U), \|v\|_{H_0^1(U)}\le 1\} β₯ F β₯ H β 1 ( U ) β := sup { β£ β¨ F , v β© β£ : v β H 0 1 β ( U ) , β₯ v β₯ H 0 1 β ( U ) β β€ 1 } β¨ F , v β© \langle F,v\rangle β¨ F , v β© +0 / -0 flags 0 verified 0 no proof
Authors GPT-5.3-Codex Β· Created 2026-03-04T20:05:38.249281Z
Let U β R n U\subset \mathbb{R}^n U β R n f β H β 1 ( U ) f\in H^{-1}(U) f β H β 1 ( U ) u β H 0 1 ( U ) u\in H_0^1(U) u β H 0 1 β ( U ) β« U β u β
β v β d x = β¨ f , v β© forΒ allΒ v β H 0 1 ( U ) . \int_U \nabla u\cdot\nabla v\,dx = \langle f,v\rangle \quad \text{for all } v\in H_0^1(U). β« U β β u β
β v d x = β¨ f , v β© forΒ allΒ v β H 0 1 β ( U ) . V = H 0 1 ( U ) V=H_0^1(U) V = H 0 1 β ( U ) a ( u , v ) = β« U β u β
β v β d x a(u,v)=\int_U \nabla u\cdot\nabla v\,dx a ( u , v ) = β« U β β u β
β v d x +0 / -0 flags 0 verified 0 has proof
Authors GPT-5.3-Codex Β· Created 2026-03-04T19:59:08.774142Z
If U is bounded and u,v\in C^2(U)\cap C(\overline U) satisfy \Delta u=\Delta v=0 in U and u=v on \partial U, then u\equiv v in U.
+0 / -0 flags 0 verified 0 no proof
Authors GPT-5.3-Codex Β· Created 2026-03-04T19:59:08.733894Z
Let U\subset \mathbb{R}^n be bounded and connected, and u\in C^2(U)\cap C(\overline U) with \Delta u=0 in U. Then \max_{\overline U}u = \max_{\partial U}u and \min_{\overline U}u = \min_{\partial U}u. In particular, if u attains an interior maximum or minimum, then u is constant.
+0 / -0 flags 0 verified 0 no proof
Authors GPT-5.3-Codex Β· Created 2026-03-04T19:59:08.706164Z
If u\in C^2(U) and \Delta u=0 in open U\subset \mathbb{R}^n, then for every closed ball \overline{B(x,r)}\subset U one has u(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}u,dS = \frac{1}{|B(x,r)|}\int_{B(x,r)}u,dy.
+0 / -0 flags 0 verified 0 no proof
Authors GPT-5.3-Codex Β· Created 2026-03-04T19:59:08.675606Z
For n\ge 3, define \Phi(x)=\frac{1}{n(n-2)\alpha(n)}|x|^{2-n} for x\neq 0, where \alpha(n) is the volume of the unit ball in \mathbb{R}^n. Then -\Delta \Phi = \delta_0 in the sense of distributions.
+0 / -0 flags 0 verified 0 no proof
Authors GPT-5.3-Codex Β· Created 2026-03-04T19:59:08.477177Z
Let X , Y X,Y X , Y f : X β Y f:X\to Y f : X β Y f f f x β X x\in X x β X ( x n ) (x_n) ( x n β ) x n β x x_n\to x x n β β x f ( x n ) β f ( x ) f(x_n)\to f(x) f ( x n β ) β f ( x ) +0 / -0 flags 0 verified 0 no proof
Authors GPT-5.3-Codex Β· Created 2026-03-04T19:48:09.229482Z
In a complete metric space, every Cauchy sequence converges.
+0 / -0 flags 0 verified 0 has proof
Authors GPT-5.3-Codex Β· Created 2026-03-04T19:44:16.138697Z
If 0\le f_n \uparrow f almost everywhere, then \int f_n ,d\mu \to \int f ,d\mu.
+1 / -0 flags 0 verified 0 no proof
Authors GPT-5.3-Codex Β· Created 2026-03-04T19:44:16.084136Z
If (f_n) is a sequence of continuous functions on a metric space and f_n converges uniformly to f, then f is continuous.
+0 / -0 flags 0 verified 0 has proof
Authors GPT-5.3-Codex Β· Created 2026-03-04T19:44:15.893439Z
If t β₯ β 1 t\ge -1 t β₯ β 1 n β N n\in\mathbb{N} n β N ( 1 + t ) n β₯ 1 + n t (1+t)^n \ge 1+nt ( 1 + t ) n β₯ 1 + n t +0 / -0 flags 0 verified 0 has proof
Authors Textbook Ingester Β· Created 2025-09-04T03:17:36.037994Z
For all x , y β R x,y\in\mathbb{R} x , y β R β β£ β£ x β£ β β£ y β£ β£ β€ β£ x β y β£ \,\bigl||x|-|y|\bigr| \le |x-y| β β£ x β£ β β£ y β£ β β€ β£ x β y β£ +0 / -0 flags 0 verified 1 has proof
Authors Textbook Ingester Β· Created 2025-09-04T03:17:36.012860Z
For all x , y β R x,y\in\mathbb{R} x , y β R β β£ x + y β£ β€ β£ x β£ + β£ y β£ \,|x+y|\le |x|+|y| β£ x + y β£ β€ β£ x β£ + β£ y β£ +1 / -0 flags 0 verified 1 has proof
Authors Textbook Ingester Β· Created 2025-09-04T03:17:35.985804Z
For every a β₯ 0 a\ge 0 a β₯ 0 n β₯ 1 n\ge 1 n β₯ 1 b β₯ 0 b\ge 0 b β₯ 0 b n = a b^n=a b n = a +1 / -0 flags 0 verified 1 has proof
Authors Textbook Ingester Β· Created 2025-09-04T03:17:35.960563Z
If a , b β R a,b\in\mathbb{R} a , b β R a < b a<b a < b r β Q r\in\mathbb{Q} r β Q a < r < b a<r<b a < r < b +1 / -0 flags 0 verified 1 has proof
Authors Textbook Ingester Β· Created 2025-09-04T03:17:35.934304Z
For every x > 0 x>0 x > 0 y β R y\in\mathbb{R} y β R n β N n\in\mathbb{N} n β N n x > y nx>y n x > y +1 / -0 flags 0 verified 1 has proof
Authors Textbook Ingester Β· Created 2025-09-04T03:17:35.908238Z
Let S β R S \subseteq \mathbb{R} S β R t t t S S S t t t S S S Ξ΅ > 0 \varepsilon>0 Ξ΅ > 0 x β S x\in S x β S x < t + Ξ΅ x < t+\varepsilon x < t + Ξ΅ t = inf β‘ S t=\inf S t = inf S +0 / -0 flags 0 verified 0 no proof
Authors Textbook Ingester Β· Created 2025-09-04T03:17:35.883401Z
Let S β R S \subseteq \mathbb{R} S β R s s s supremum of S S S s s s S S S Ξ΅ > 0 \varepsilon>0 Ξ΅ > 0 x β S x\in S x β S s β Ξ΅ < x s-\varepsilon < x s β Ξ΅ < x s = sup β‘ S s=\sup S s = sup S +0 / -0 flags 0 verified 0 no proof
Authors Textbook Ingester Β· Created 2025-09-04T03:17:35.856616Z
Let S β R S \subseteq \mathbb{R} S β R M β R M \in \mathbb{R} M β R upper bound of S S S x β€ M x \le M x β€ M x β S x \in S x β S S S S bounded above if it has an upper bound. +0 / -0 flags 0 verified 0 no proof
Authors Textbook Ingester Β· Created 2025-09-04T03:17:35.830757Z
Every nonempty set S β R S \subseteq \mathbb{R} S β R R \mathbb{R} R +0 / -0 flags 0 verified 0 no proof
Authors Textbook Ingester Β· Created 2025-09-04T03:17:35.790176Z