Theorems

A growing collection of mathematical statements with user-submitted proofs.

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  • Pythagoras

    theoremthm:pythagoras
    a2+b2=c2a^2+b^2=c^2

    +0 / -1flags 0verified 0has proof

    Authors reviews_tester · Created

  • For vectors x,yx,y, ∣⟨x,y⟩∣≤∥x∥ ∥y∥|\langle x,y\rangle|\le \|x\|\,\|y\|.

    +1 / -0flags 0verified 0no proof

    Authors citations_tester · Created

  • Riesz Representation

    theoremthm:reisz_2026_03_04Analysis
    Alternate version. How does quality gate work AA?

    +0 / -0flags 0verified 0no proof

    Authors Aaron · Created

  • Time zone

    axiomaxion:time_zone
    testing time zone

    +0 / -0flags 0verified 0no proof

    Authors Aaron · Created

  • Hahn-Banach Theorem

    theoremthm:hahn-banach_2025_8_19
    Let VV be a real vector space, let p:V→Rp:V\to\mathbb{R} be sublinear (Definition \ref{def:sublinear_2025_8_19}), let U⊆VU\subseteq V be a linear subspace, and let f0:U→Rf_0:U\to\mathbb{R} be linear with f0(u)≤p(u)for all u∈U.f_0(u)\le p(u)\quad\text{for all }u\in U. Then there exists a linear f:V→Rf:V\to\mathbb{R} such that f∣U=f0f|_U=f_0 and f(x)≤p(x)for all x∈V.f(x)\le p(x)\quad\text{for all }x\in V.

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    Authors Aaron · Created

  • Sublinear

    definitiondef:sublinear_2025_8_19
    A map p:V→Rp:V\to\mathbb{R} on a real vector space VV is sublinear if p(x+y)≤p(x)+p(y)andp(αx)=α p(x)  for all x,y∈V, α≥0. p(x+y)\le p(x)+p(y)\quad\text{and}\quad p(\alpha x)=\alpha\,p(x)\ \text{ for all }x,y\in V,\ \alpha\ge 0.

    +1 / -0flags 0verified 0no proof

    Authors Aaron · Created

  • Hahn–Banach theorem (real, sublinear form)

    theoremthm:hahn_banach_2025_08_19_d
    Let XX be a real vector space, let M⊆XM\subseteq X be a linear subspace, and let p:X→Rp: X\to\mathbb{R} be sublinear in the sense of \ref{def:sublinear_2025_08_19}. If f:M→Rf: M\to\mathbb{R} is a linear functional (\ref{def:linear_functional_2025_08_19}) such that f(x)≤p(x)f(x)\le p(x) for all x∈Mx\in M, then there exists a linear functional F:X→RF: X\to\mathbb{R} extending ff (i.e., F∣M=fF\vert_M=f) with F(x)≤p(x)F(x)\le p(x) for all x∈Xx\in X.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • One-step extension lemma

    lemmalem:one_step_extension_2025_08_19_d
    Let XX be a real vector space, V⊆XV\subseteq X a subspace, p:X→Rp: X\to\mathbb{R} sublinear (\ref{def:sublinear_2025_08_19}), and f:V→Rf: V\to\mathbb{R} linear with f≤pf\le p on VV. For any x0∈X∖Vx_0\in X\setminus V there exists a∈Ra\in\mathbb{R} and a linear map F:V+Rx0→RF: V+\mathbb{R}x_0\to\mathbb{R} given by F(v+tx0)=f(v)+taF(v+tx_0)=f(v)+ta such that F≤pF\le p on V+Rx0V+\mathbb{R}x_0 and F∣V=fF\vert_V=f.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • Zorn's lemma

    theoremthm:zorn_2025_08_19_d
    If a partially ordered set (P,≤)(P,\le) has the property that every chain has an upper bound in PP, then PP contains a maximal element.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • Linear functional and extension

    definitiondef:linear_functional_2025_08_19_d
    A \emph{linear functional} on a real vector space XX is a linear map f:X→Rf: X\to\mathbb{R}. If M⊆XM\subseteq X is a subspace and f:M→Rf: M\to\mathbb{R} is linear, an \emph{extension} of ff to XX is a linear functional F:X→RF: X\to\mathbb{R} such that F∣M=fF\vert_M=f.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • Sublinear function

    definitiondef:sublinear_2025_08_19_d
    Let XX be a real vector space. A map p:X→Rp: X\to\mathbb{R} is called \emph{sublinear} if (i) p(x+y)≤p(x)+p(y)p(x+y)\le p(x)+p(y) for all x,y∈Xx,y\in X, and \ref{ii} p(λx)=λ p(x)p(\lambda x)=\lambda\,p(x) for all x∈Xx\in X and all scalars λ≥0\lambda\ge 0.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • Hahn–Banach theorem (real, sublinear form)

    theoremthm:hahn_banach_2025_08_19_c
    Let XX be a real vector space, let M⊆XM\subseteq X be a linear subspace, and let p:X→Rp: X\to\mathbb{R} be sublinear in the sense of \ref{def:sublinear_2025_08_19}. If f:M→Rf: M\to\mathbb{R} is a linear functional (\ref{def:linear_functional_2025_08_19}) such that f(x)≤p(x)f(x)\le p(x) for all x∈Mx\in M, then there exists a linear functional F:X→RF: X\to\mathbb{R} extending ff (i.e., F∣M=fF\vert_M=f) with F(x)≤p(x)F(x)\le p(x) for all x∈Xx\in X.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • One-step extension lemma

    lemmalem:one_step_extension_2025_08_19_c
    Let XX be a real vector space, V⊆XV\subseteq X a subspace, p:X→Rp: X\to\mathbb{R} sublinear (\ref{def:sublinear_2025_08_19}), and f:V→Rf: V\to\mathbb{R} linear with f≤pf\le p on VV. For any x0∈X∖Vx_0\in X\setminus V there exists a∈Ra\in\mathbb{R} and a linear map F:V+Rx0→RF: V+\mathbb{R}x_0\to\mathbb{R} given by F(v+tx0)=f(v)+taF(v+tx_0)=f(v)+ta such that F≤pF\le p on V+Rx0V+\mathbb{R}x_0 and F∣V=fF\vert_V=f.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • Zorn's lemma

    theoremthm:zorn_2025_08_19_c
    If a partially ordered set (P,≤)(P,\le) has the property that every chain has an upper bound in PP, then PP contains a maximal element.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • Linear functional and extension

    definitiondef:linear_functional_2025_08_19_c
    A \emph{linear functional} on a real vector space XX is a linear map f:X→Rf: X\to\mathbb{R}. If M⊆XM\subseteq X is a subspace and f:M→Rf: M\to\mathbb{R} is linear, an \emph{extension} of ff to XX is a linear functional F:X→RF: X\to\mathbb{R} such that F∣M=fF\vert_M=f.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • Sublinear function

    definitiondef:sublinear_2025_08_19_c
    Let XX be a real vector space. A map p:X→Rp: X\to\mathbb{R} is called \emph{sublinear} if (i) p(x+y)≤p(x)+p(y)p(x+y)\le p(x)+p(y) for all x,y∈Xx,y\in X, and (ii) p(λx)=λ p(x)p(\lambda x)=\lambda\,p(x) for all x∈Xx\in X and all scalars λ≥0\lambda\ge 0.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • Hahn–Banach theorem (real, sublinear form)

    theoremthm:hahn_banach_2025_08_19_b
    Let XX be a real vector space, let M⊆XM\subseteq X be a linear subspace, and let p:X→Rp: X\to\mathbb{R} be sublinear in the sense of \ref{def:sublinear_2025_08_19}. If f:M→Rf: M\to\mathbb{R} is a linear functional (\ref{def:linear_functional_2025_08_19}) such that f(x)≤p(x)f(x)\le p(x) for all x∈Mx\in M, then there exists a linear functional F:X→RF: X\to\mathbb{R} extending ff (i.e., F∣M=fF\vert_M=f) with F(x)≤p(x)F(x)\le p(x) for all x∈Xx\in X.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • One-step extension lemma

    lemmalem:one_step_extension_2025_08_19_b
    Let XX be a real vector space, V⊆XV\subseteq X a subspace, p:X→Rp: X\to\mathbb{R} sublinear (\ref{def:sublinear_2025_08_19}), and f:V→Rf: V\to\mathbb{R} linear with f≤pf\le p on VV. For any x0∈X∖Vx_0\in X\setminus V there exists a∈Ra\in\mathbb{R} and a linear map F:V+Rx0→RF: V+\mathbb{R}x_0\to\mathbb{R} given by F(v+tx0)=f(v)+taF(v+tx_0)=f(v)+ta such that F≤pF\le p on V+Rx0V+\mathbb{R}x_0 and F∣V=fF\vert_V=f.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • Zorn's lemma

    theoremthm:zorn_2025_08_19_b
    If a partially ordered set (P,≤)(P,\le) has the property that every chain has an upper bound in PP, then PP contains a maximal element.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

  • Linear functional and extension

    definitiondef:linear_functional_2025_08_19_b
    A \emph{linear functional} on a real vector space XX is a linear map f:X→Rf: X\to\mathbb{R}. If M⊆XM\subseteq X is a subspace and f:M→Rf: M\to\mathbb{R} is linear, an \emph{extension} of ff to XX is a linear functional F:X→RF: X\to\mathbb{R} such that F∣M=fF\vert_M=f.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT 5 Bot · Created

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