id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-22600 | Inverse Laplace Transform of s^2 over s^3 - a^3 | :$\map {\laptrans {\dfrac 1 3 \paren {e^{a t} + 2 e^{-a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } } s = \dfrac {s^2} {s^3 - a^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 3 \paren {e^{a t} + 2 e^{-a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } } s
| r = \dfrac 1 3 \map {\laptrans {e^{a t} } } s + \dfrac 2 3 \map {\laptrans {e^{-a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } s
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r =... | :$\map {\laptrans {\dfrac 1 3 \paren {e^{a t} + 2 e^{-a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } } s = \dfrac {s^2} {s^3 - a^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 3 \paren {e^{a t} + 2 e^{-a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } } s
| r = \dfrac 1 3 \map {\laptrans {e^{a t} } } s + \dfrac 2 3 \map {\laptrans {e^{-a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } s
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn |... | Inverse Laplace Transform of s^2 over s^3 - a^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^2_over_s^3_-_a^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^2_over_s^3_-_a^3 | [
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Exponential Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential times Cosine",
"Laplace Transform of Exponential",
"Definition:Common Denominator",
"Difference of Two Powers/Examples/Difference of Two Cubes"
] |
proofwiki-22601 | Inverse Laplace Transform of 1 over s^4 + 4 a^4 | :$\map {\laptrans {\dfrac 1 {4 a^3} \paren {\sin a t \cosh a t - \cos a t \sinh a t} } } s = \dfrac 1 {s^4 + 4 a^4}$ | First we note that:
{{begin-eqn}}
{{eqn | l = \sin a t \cosh a t - \cos a t \sinh a t
| r = \sin a t \paren {\dfrac {e^{a t} + e^{-a t} } 2} - \cos a t \paren {\dfrac {e^{a t} - e^{-a t} } 2}
| c = {{Defof|Hyperbolic Cosine}} and {{Defof|Hyperbolic Sine}}
}}
{{eqn | n = 1
| r = \dfrac 1 2 \paren {e^{a... | :$\map {\laptrans {\dfrac 1 {4 a^3} \paren {\sin a t \cosh a t - \cos a t \sinh a t} } } s = \dfrac 1 {s^4 + 4 a^4}$ | First we note that:
{{begin-eqn}}
{{eqn | l = \sin a t \cosh a t - \cos a t \sinh a t
| r = \sin a t \paren {\dfrac {e^{a t} + e^{-a t} } 2} - \cos a t \paren {\dfrac {e^{a t} - e^{-a t} } 2}
| c = {{Defof|Hyperbolic Cosine}} and {{Defof|Hyperbolic Sine}}
}}
{{eqn | n = 1
| r = \dfrac 1 2 \paren {e^{... | Inverse Laplace Transform of 1 over s^4 + 4 a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_1_over_s^4_+_4_a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_1_over_s^4_+_4_a^4 | [
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Exponential Function",
"Inverse Laplace Transforms of Rational Fun... | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential times Sine",
"Laplace Transform of Exponential times Cosine",
"Definition:Common Denominator",
"Sophie Germain's Identity"
] |
proofwiki-22602 | Inverse Laplace Transform of s over s^4 + 4 a^4 | :$\map {\laptrans {\dfrac {\sin a t \sinh a t} {2 a^2} } } s = \dfrac s {s^4 + 4 a^4}$ | First we note that:
{{begin-eqn}}
{{eqn | l = \sin a t \sinh a t
| r = \sin a t \paren {\dfrac {e^{a t} - e^{-a t} } 2}
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | n = 1
| r = \dfrac 1 2 \paren {e^{a t} \sin a t - e^{-a t} \sin a t}
| c =
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \map {\la... | :$\map {\laptrans {\dfrac {\sin a t \sinh a t} {2 a^2} } } s = \dfrac s {s^4 + 4 a^4}$ | First we note that:
{{begin-eqn}}
{{eqn | l = \sin a t \sinh a t
| r = \sin a t \paren {\dfrac {e^{a t} - e^{-a t} } 2}
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | n = 1
| r = \dfrac 1 2 \paren {e^{a t} \sin a t - e^{-a t} \sin a t}
| c =
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \map ... | Inverse Laplace Transform of s over s^4 + 4 a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s_over_s^4_+_4_a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s_over_s^4_+_4_a^4 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Exponential Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential times Sine",
"Definition:Common Denominator",
"Sophie Germain's Identity"
] |
proofwiki-22603 | Inverse Laplace Transform of s^2 over s^4 + 4 a^4 | :$\map {\laptrans {\dfrac 1 {2 a} \paren {\sin a t \cosh a t + \cos a t \sinh a t} } } s = \dfrac {s^2} {s^4 + 4 a^4}$ | First we note that:
{{begin-eqn}}
{{eqn | l = \sin a t \cosh a t + \cos a t \sinh a t
| r = \sin a t \paren {\dfrac {e^{a t} + e^{-a t} } 2} + \cos a t \paren {\dfrac {e^{a t} - e^{-a t} } 2}
| c = {{Defof|Hyperbolic Cosine}} and {{Defof|Hyperbolic Sine}}
}}
{{eqn | n = 1
| r = \dfrac 1 2 \paren {e^{a... | :$\map {\laptrans {\dfrac 1 {2 a} \paren {\sin a t \cosh a t + \cos a t \sinh a t} } } s = \dfrac {s^2} {s^4 + 4 a^4}$ | First we note that:
{{begin-eqn}}
{{eqn | l = \sin a t \cosh a t + \cos a t \sinh a t
| r = \sin a t \paren {\dfrac {e^{a t} + e^{-a t} } 2} + \cos a t \paren {\dfrac {e^{a t} - e^{-a t} } 2}
| c = {{Defof|Hyperbolic Cosine}} and {{Defof|Hyperbolic Sine}}
}}
{{eqn | n = 1
| r = \dfrac 1 2 \paren {e^{... | Inverse Laplace Transform of s^2 over s^4 + 4 a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^2_over_s^4_+_4_a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^2_over_s^4_+_4_a^4 | [
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Exponential Function",
"Inverse Laplace Transforms of Rational Fun... | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential times Sine",
"Laplace Transform of Exponential times Cosine",
"Definition:Common Denominator",
"Sophie Germain's Identity"
] |
proofwiki-22604 | Laplace Transform of cos a t by cosh a t | :$\map {\laptrans {\cos a t \cosh a t} } s = \dfrac {s^3} {s^4 + 4 a^4}$ | First we note that:
{{begin-eqn}}
{{eqn | l = \cos a t \cosh a t
| r = \cos a t \paren {\dfrac {e^{a t} + e^{-a t} } 2}
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | n = 1
| r = \dfrac 1 2 \paren {e^{a t} \cos a t + e^{-a t} \cos a t}
| c =
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \map {\... | :$\map {\laptrans {\cos a t \cosh a t} } s = \dfrac {s^3} {s^4 + 4 a^4}$ | First we note that:
{{begin-eqn}}
{{eqn | l = \cos a t \cosh a t
| r = \cos a t \paren {\dfrac {e^{a t} + e^{-a t} } 2}
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | n = 1
| r = \dfrac 1 2 \paren {e^{a t} \cos a t + e^{-a t} \cos a t}
| c =
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \ma... | Laplace Transform of cos a t by cosh a t | https://proofwiki.org/wiki/Laplace_Transform_of_cos_a_t_by_cosh_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_cos_a_t_by_cosh_a_t | [
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Laplace Transforms involving Exponential Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential times Cosine",
"Definition:Common Denominator",
"Sophie Germain's Identity"
] |
proofwiki-22605 | Inverse Laplace Transform of 1 over s^4 - a^4 | :$\map {\laptrans {\dfrac 1 {2 a^3} \paren {\sinh a t - \sin a t} } } s = \dfrac 1 {s^4 - a^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 {2 a^3} \paren {\sinh a t - \sin a t} } } s
| r = \dfrac 1 {2 a^3} \paren {\map {\laptrans {\sinh a t} } s - \map {\laptrans {\sin a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {2 a^3} \paren {\dfrac a {s^2 - a^2} - \dfra... | :$\map {\laptrans {\dfrac 1 {2 a^3} \paren {\sinh a t - \sin a t} } } s = \dfrac 1 {s^4 - a^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 {2 a^3} \paren {\sinh a t - \sin a t} } } s
| r = \dfrac 1 {2 a^3} \paren {\map {\laptrans {\sinh a t} } s - \map {\laptrans {\sin a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {2 a^3} \paren {\dfrac a {s^2 - a^2} - \... | Inverse Laplace Transform of 1 over s^4 - a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_1_over_s^4_-_a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_1_over_s^4_-_a^4 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Hyperbolic Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Hyperbolic Sine",
"Laplace Transform of Sine",
"Definition:Common Denominator",
"Difference of Two Squares"
] |
proofwiki-22606 | Inverse Laplace Transform of s over s^4 - a^4 | :$\map {\laptrans {\dfrac 1 {2 a^2} \paren {\cosh a t - \cos a t} } } s = \dfrac s {s^4 - a^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 {2 a^2} \paren {\cosh a t - \cos a t} } } s
| r = \dfrac 1 {2 a^2} \paren {\map {\laptrans {\cosh a t} } s - \map {\laptrans {\cos a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {2 a^2} \paren {\dfrac s {s^2 - a^2} - \dfra... | :$\map {\laptrans {\dfrac 1 {2 a^2} \paren {\cosh a t - \cos a t} } } s = \dfrac s {s^4 - a^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 {2 a^2} \paren {\cosh a t - \cos a t} } } s
| r = \dfrac 1 {2 a^2} \paren {\map {\laptrans {\cosh a t} } s - \map {\laptrans {\cos a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {2 a^2} \paren {\dfrac s {s^2 - a^2} - \... | Inverse Laplace Transform of s over s^4 - a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s_over_s^4_-_a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s_over_s^4_-_a^4 | [
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Hyperbolic Cosine",
"Laplace Transform of Cosine",
"Definition:Common Denominator",
"Difference of Two Squares"
] |
proofwiki-22607 | Inverse Laplace Transform of s^2 over s^4 - a^4 | :$\map {\laptrans {\dfrac 1 {2 a} \paren {\sinh a t + \sin a t} } } s = \dfrac {s^2} {s^4 - a^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 {2 a} \paren {\sinh a t + \sin a t} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {\sinh a t} } s + \map {\laptrans {\sin a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {2 a} \paren {\dfrac a {s^2 - a^2} + \dfrac a {s... | :$\map {\laptrans {\dfrac 1 {2 a} \paren {\sinh a t + \sin a t} } } s = \dfrac {s^2} {s^4 - a^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 {2 a} \paren {\sinh a t + \sin a t} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {\sinh a t} } s + \map {\laptrans {\sin a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {2 a} \paren {\dfrac a {s^2 - a^2} + \dfrac ... | Inverse Laplace Transform of s^2 over s^4 - a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^2_over_s^4_-_a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^2_over_s^4_-_a^4 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Hyperbolic Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Hyperbolic Sine",
"Laplace Transform of Sine",
"Definition:Common Denominator",
"Difference of Two Squares"
] |
proofwiki-22608 | Inverse Laplace Transform of s^3 over s^4 - a^4 | :$\map {\laptrans {\dfrac 1 2 \paren {\cosh a t - \cos a t} } } s = \dfrac {s^3} {s^4 - a^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 2 \paren {\cosh a t + \cos a t} } } s
| r = \dfrac 1 2 \paren {\map {\laptrans {\cosh a t} } s + \map {\laptrans {\cos a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 2 \paren {\dfrac s {s^2 - a^2} + \dfrac s {s^2 + a^2} }
... | :$\map {\laptrans {\dfrac 1 2 \paren {\cosh a t - \cos a t} } } s = \dfrac {s^3} {s^4 - a^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 2 \paren {\cosh a t + \cos a t} } } s
| r = \dfrac 1 2 \paren {\map {\laptrans {\cosh a t} } s + \map {\laptrans {\cos a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 2 \paren {\dfrac s {s^2 - a^2} + \dfrac s {s^2 + a^2... | Inverse Laplace Transform of s^3 over s^4 - a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^3_over_s^4_-_a^4 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^3_over_s^4_-_a^4 | [
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Hyperbolic Cosine",
"Laplace Transform of Cosine",
"Definition:Common Denominator",
"Difference of Two Squares"
] |
proofwiki-22609 | Equivalence of Definitions of Complete Lattice Isomorphism | Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be complete lattices.
Let $\phi: A_1 \to A_2$ be a mapping.
{{TFAE|def = Complete Lattice Isomorphism}}
=== Definition 1 ===
{{:Definition:Complete Lattice Isomorphism/Definition 1}}
=== Definition 2 ===
{{:Definition:Complete Lattice Isomorphism... | === Definition 1 implies Definition 2 ===
Let $\phi : L_1 \to L_2$ be a bijective complete lattice homomorphism.
{{:Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 2}}{{qed|lemma}} | Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be [[Definition:Complete Lattice|complete lattices]].
Let $\phi: A_1 \to A_2$ be a [[Definition:Mapping|mapping]].
{{TFAE|def = Complete Lattice Isomorphism}}
=== [[Definition:Complete Lattice Isomorphism/Definition 1|Definition 1]] ===
{{:De... | === [[Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 2|Definition 1 implies Definition 2]] ===
Let $\phi : L_1 \to L_2$ be a [[Definition:Bijection|bijective]] [[Definition:Complete Lattice Homomorphism|complete lattice homomorphism]].
{{:Equivalence of Definitions of Comple... | Equivalence of Definitions of Complete Lattice Isomorphism | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complete_Lattice_Isomorphism | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complete_Lattice_Isomorphism | [
"Equivalence of Definitions of Complete Lattice Isomorphism",
"Complete Lattice Isomorphisms"
] | [
"Definition:Complete Lattice",
"Definition:Mapping",
"Definition:Complete Lattice Isomorphism/Definition 1",
"Definition:Complete Lattice Isomorphism/Definition 2",
"Definition:Complete Lattice Isomorphism/Definition 3"
] | [
"Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 2",
"Definition:Bijection",
"Definition:Complete Lattice Homomorphism"
] |
proofwiki-22610 | Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 2 | Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be complete lattices.
Let $\phi : L_1 \to L_2$ be a bijective complete lattice homomorphism.
Then:
:$\phi : L_1 \to L_2$ is an order isomorphism | From Complete Lattice Homomorphism is Lattice Homomorphism:
:$\phi : \struct {A_1, \vee_1, \wedge_1, \preceq_1} \to \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ is a lattice homomorphism
where
:$\wedge_1, \vee_1$ denote the meet and join on $L_1$
:$\wedge_2, \vee_2$ denote the meet and join on $L_2$
Hence $\phi : \st... | Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be [[Definition:Complete Lattice|complete lattices]].
Let $\phi : L_1 \to L_2$ be a [[Definition:Bijection|bijective]] [[Definition:Complete Lattice Homomorphism|complete lattice homomorphism]].
Then:
:$\phi : L_1 \to L_2$ is an [[Definition:O... | From [[Complete Lattice Homomorphism is Lattice Homomorphism]]:
:$\phi : \struct {A_1, \vee_1, \wedge_1, \preceq_1} \to \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ is a [[Definition:Lattice Homomorphism|lattice homomorphism]]
where
:$\wedge_1, \vee_1$ denote the [[Definition:Meet|meet]] and [[Definition:Join|join]] o... | Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complete_Lattice_Isomorphism/Definition_1_Implies_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complete_Lattice_Isomorphism/Definition_1_Implies_Definition_2 | [
"Equivalence of Definitions of Complete Lattice Isomorphism"
] | [
"Definition:Complete Lattice",
"Definition:Bijection",
"Definition:Complete Lattice Homomorphism",
"Definition:Order Isomorphism"
] | [
"Complete Lattice Homomorphism is Lattice Homomorphism",
"Definition:Lattice Homomorphism",
"Definition:Meet",
"Definition:Join",
"Definition:Meet",
"Definition:Join",
"Definition:Bijection",
"Definition:Lattice Homomorphism",
"Definition:Lattice Isomorphism",
"Equivalence of Definitions of Lattic... |
proofwiki-22611 | Equivalence of Definitions of Complete Lattice Isomorphism/Definition 2 Implies Definition 3 | Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be complete lattices.
Let $\phi : L_1 \to L_2$ be an order isomorphism.
Then:
:$\phi : \struct {A_1, \vee_1, \wedge_1, \preceq_1} \to \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ is a lattice isomorphism
where
:$\wedge_1, \vee_1$ denote the meet... | From Complete Lattice is Lattice:
:$\struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $\struct {A_2, \vee_2, \wedge_2, \preceq_2}$ are lattices
where
:$\wedge_1, \vee_1$ denote the meet and join on $L_1$
:$\wedge_2, \vee_2$ denote the meet and join on $L_2$
Hence $\phi : \struct {A_1, \vee_1, \wedge_1, \preceq_1} \to ... | Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be [[Definition:Complete Lattice|complete lattices]].
Let $\phi : L_1 \to L_2$ be an [[Definition:Order Isomorphism|order isomorphism]].
Then:
:$\phi : \struct {A_1, \vee_1, \wedge_1, \preceq_1} \to \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ ... | From [[Complete Lattice is Lattice]]:
:$\struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $\struct {A_2, \vee_2, \wedge_2, \preceq_2}$ are [[Definition:Lattice (Order Theory)|lattices]]
where
:$\wedge_1, \vee_1$ denote the [[Definition:Meet|meet]] and [[Definition:Join|join]] on $L_1$
:$\wedge_2, \vee_2$ denote the [[... | Equivalence of Definitions of Complete Lattice Isomorphism/Definition 2 Implies Definition 3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complete_Lattice_Isomorphism/Definition_2_Implies_Definition_3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complete_Lattice_Isomorphism/Definition_2_Implies_Definition_3 | [
"Equivalence of Definitions of Complete Lattice Isomorphism"
] | [
"Definition:Complete Lattice",
"Definition:Order Isomorphism",
"Definition:Lattice Isomorphism",
"Definition:Meet",
"Definition:Join",
"Definition:Meet",
"Definition:Join"
] | [
"Complete Lattice is Lattice",
"Definition:Lattice (Order Theory)",
"Definition:Meet",
"Definition:Join",
"Definition:Meet",
"Definition:Join",
"Definition:Order Isomorphism",
"Definition:Lattice (Order Theory)",
"definition:Lattice Isomorphism",
"Definition:Lattice Isomorphism"
] |
proofwiki-22612 | Equivalence of Definitions of Complete Lattice Isomorphism/Definition 3 Implies Definition 1 | Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be complete lattices.
Let $\phi : \struct {A_1, \vee_1, \wedge_1, \preceq_1} \to \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be a lattice isomorphism
where
:$\wedge_1, \vee_1$ denote the meet and join on $L_1$
:$\wedge_2, \vee_2$ denote the me... | From Complete Lattice is Lattice:
:$\struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $\struct {A_2, \vee_2, \wedge_2, \preceq_2}$ are lattices
where
:$\wedge_1, \vee_1$ denote the meet and join on $L_1$
:$\wedge_2, \vee_2$ denote the meet and join on $L_2$
By definition of lattice isomorphism:
:$\phi : L_1 \to L_2$ i... | Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be [[Definition:Complete Lattice|complete lattices]].
Let $\phi : \struct {A_1, \vee_1, \wedge_1, \preceq_1} \to \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be a [[Definition:Lattice Isomorphism|lattice isomorphism]]
where
:$\wedge_1, \vee_1$ ... | From [[Complete Lattice is Lattice]]:
:$\struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $\struct {A_2, \vee_2, \wedge_2, \preceq_2}$ are [[Definition:Lattice (Order Theory)|lattices]]
where
:$\wedge_1, \vee_1$ denote the [[Definition:Meet|meet]] and [[Definition:Join|join]] on $L_1$
:$\wedge_2, \vee_2$ denote the [[... | Equivalence of Definitions of Complete Lattice Isomorphism/Definition 3 Implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complete_Lattice_Isomorphism/Definition_3_Implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complete_Lattice_Isomorphism/Definition_3_Implies_Definition_1 | [
"Equivalence of Definitions of Complete Lattice Isomorphism"
] | [
"Definition:Complete Lattice",
"Definition:Lattice Isomorphism",
"Definition:Meet",
"Definition:Join",
"Definition:Meet",
"Definition:Join",
"Definition:Bijection",
"Definition:Complete Lattice Homomorphism"
] | [
"Complete Lattice is Lattice",
"Definition:Lattice (Order Theory)",
"Definition:Meet",
"Definition:Join",
"Definition:Meet",
"Definition:Join",
"Definition:Lattice Isomorphism",
"Definition:Order Isomorphism",
"Order Isomorphism Preserves Infima and Suprema",
"Definition:Arbitrary Join Preserving ... |
proofwiki-22613 | Complete Lattice Homomorphism is Lattice Homomorphism | Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be complete lattices.
Let $\phi : L_1 \to L_2$ be a complete lattice homomorphism.
Then:
:$\phi : \struct {S_1, \vee_1, \wedge_1, \preceq_1} \to \struct {S_2, \vee_2, \wedge_2, \preceq_2}$ is a lattice homomorphism
where:
:$\wedge_1, \vee_1$ den... | From Complete Lattice is Lattice:
:$\struct {S_1, \vee_1, \wedge_1, \preceq_1}$ and $\struct {S_2, \vee_2, \wedge_2, \preceq_2}$ are lattices
where
:$\wedge_1, \vee_1$ denote the meet and join on $L_1$
:$\wedge_2, \vee_2$ denote the meet and join on $L_2$
By definition of complete lattice homomorphsim:
:$\phi$ satis... | Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be [[Definition:Complete Lattice|complete lattices]].
Let $\phi : L_1 \to L_2$ be a [[Definition:Complete Lattice Homomorphism|complete lattice homomorphism]].
Then:
:$\phi : \struct {S_1, \vee_1, \wedge_1, \preceq_1} \to \struct {S_2, \vee_2,... | From [[Complete Lattice is Lattice]]:
:$\struct {S_1, \vee_1, \wedge_1, \preceq_1}$ and $\struct {S_2, \vee_2, \wedge_2, \preceq_2}$ are [[Definition:Lattice (Order Theory)|lattices]]
where
:$\wedge_1, \vee_1$ denote the [[Definition:Meet|meet]] and [[Definition:Join|join]] on $L_1$
:$\wedge_2, \vee_2$ denote the [[... | Complete Lattice Homomorphism is Lattice Homomorphism | https://proofwiki.org/wiki/Complete_Lattice_Homomorphism_is_Lattice_Homomorphism | https://proofwiki.org/wiki/Complete_Lattice_Homomorphism_is_Lattice_Homomorphism | [
"Complete Lattice Homomorphisms",
"Lattice Homomorphisms"
] | [
"Definition:Complete Lattice",
"Definition:Complete Lattice Homomorphism",
"Definition:Lattice Homomorphism",
"Definition:Meet",
"Definition:Join",
"Definition:Meet",
"Definition:Join"
] | [
"Complete Lattice is Lattice",
"Definition:Lattice (Order Theory)",
"Definition:Meet",
"Definition:Join",
"Definition:Meet",
"Definition:Join",
"Definition:Complete Lattice Homomorphism",
"Axiom:Complete Lattice Homomorphism Axioms",
"Definition:Meet",
"Definition:Join",
"Definition:Morphism Pro... |
proofwiki-22614 | Inverse of Complete Lattice Isomorphism is Complete Lattice Isomorphism | Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be complete lattices.
Let $\phi: L_1 \to L_2$ be a complete lattice isomorphism.
Let $\phi^{-1} : A_2 \to A_1$ be the inverse of $\phi : A_1 \to A_2$.
Then:
:$\phi^{-1} : L_2 \to L_1$ is a complete lattice isomorphism | By definition, a complete lattice isomorphism is a bijective complete lattice homomorphism.
The result follows from Inverse of Bijective Complete Lattice Homomorphism is Bijective Complete Lattice Homomorphism.
{{qed}}
Category:Complete Lattice Isomorphisms
g7m0iqfr6hqlityb0mxbmlnlreyi5sg | Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be [[Definition:Complete Lattice|complete lattices]].
Let $\phi: L_1 \to L_2$ be a [[Definition:Complete Lattice Isomorphism|complete lattice isomorphism]].
Let $\phi^{-1} : A_2 \to A_1$ be the [[Definition:Inverse Mapping|inverse]] of $\phi :... | By definition, a [[Definition:Complete Lattice Isomorphism|complete lattice isomorphism]] is a [[Definition:Bijection|bijective]] [[Definition:Complete Lattice Homomorphism|complete lattice homomorphism]].
The result follows from [[Inverse of Bijective Complete Lattice Homomorphism is Bijective Complete Lattice Homomo... | Inverse of Complete Lattice Isomorphism is Complete Lattice Isomorphism | https://proofwiki.org/wiki/Inverse_of_Complete_Lattice_Isomorphism_is_Complete_Lattice_Isomorphism | https://proofwiki.org/wiki/Inverse_of_Complete_Lattice_Isomorphism_is_Complete_Lattice_Isomorphism | [
"Complete Lattice Isomorphisms"
] | [
"Definition:Complete Lattice",
"Definition:Complete Lattice Isomorphism",
"Definition:Inverse Mapping",
"Definition:Complete Lattice Isomorphism"
] | [
"Definition:Complete Lattice Isomorphism",
"Definition:Bijection",
"Definition:Complete Lattice Homomorphism",
"Inverse of Bijective Complete Lattice Homomorphism is Bijective Complete Lattice Homomorphism",
"Category:Complete Lattice Isomorphisms"
] |
proofwiki-22615 | Category of Complete Lattices is Category | Let $\mathbf {CLat}$ denote the category of complete lattices.
Then:
:$\mathbf {CLat}$ is a metacategory | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a metacategory.
For any two complete lattice homomorphisms their composition (in the usual set theoretic sense) is again a complete lattice homomorphism by Composite Complete Lattice Homomorphisms is Complete Lattice Homomorphism.
For any complete lattice $... | Let $\mathbf {CLat}$ denote the [[Definition:Category of Complete Lattices|category of complete lattices]].
Then:
:$\mathbf {CLat}$ is a [[Definition:Metacategory|metacategory]] | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a [[Definition:Metacategory|metacategory]].
For any two [[Definition:Complete Lattice Homomorphism|complete lattice homomorphisms]] their [[Definition:Composition of Mappings|composition]] (in the usual [[Definition:Set Theory|set theoretic]] sense) is ag... | Category of Complete Lattices is Category | https://proofwiki.org/wiki/Category_of_Complete_Lattices_is_Category | https://proofwiki.org/wiki/Category_of_Complete_Lattices_is_Category | [
"Category of Complete Lattices"
] | [
"Definition:Category of Complete Lattices",
"Definition:Metacategory"
] | [
"Definition:Metacategory",
"Definition:Complete Lattice Homomorphism",
"Definition:Composition of Mappings",
"Definition:Set Theory",
"Definition:Complete Lattice Homomorphism",
"Composite Complete Lattice Homomorphisms is Complete Lattice Homomorphism",
"Definition:Complete Lattice",
"Definition:Iden... |
proofwiki-22616 | Complete Lattice Isomorphism is Isomorphism in Category CLat | Let $\mathbf{CLat}$ denote the category of complete lattices.
Let $\phi : L_1 \to L_2$ be a morphism of $\mathbf{CLat}$.
Then:
:$\phi$ is an isomorphism of $\mathbf{CLat}$ {{iff}} $\phi$ is a complete lattice isomorphsm | Let $L_1$ and $L_2$ be the complete lattices $L_1 = \struct{A_1, \preceq_1}$ and $L_2 = \struct{A_2, \preceq_2}$ respectively.
By definition of category of complete lattices:
:$\phi$ is a complete lattice homomorphisms
By definition of an isomorphism:
:$\phi$ is an isomorphism of $\mathbf{CLat}$
{{iff}}:
:$(1):$ there ... | Let $\mathbf{CLat}$ denote the [[Definition:Category of Complete Lattices|category of complete lattices]].
Let $\phi : L_1 \to L_2$ be a [[Definition:Morphism|morphism]] of $\mathbf{CLat}$.
Then:
:$\phi$ is an [[Definition:Isomorphism (Category Theory)|isomorphism]] of $\mathbf{CLat}$ {{iff}} $\phi$ is a [[Definit... | Let $L_1$ and $L_2$ be the [[Definition:Complete Lattice|complete lattices]] $L_1 = \struct{A_1, \preceq_1}$ and $L_2 = \struct{A_2, \preceq_2}$ respectively.
By definition of [[Definition:Category of Complete Lattices|category of complete lattices]]:
:$\phi$ is a [[Definition:Complete Lattice Homomorphism|complete ... | Complete Lattice Isomorphism is Isomorphism in Category CLat | https://proofwiki.org/wiki/Complete_Lattice_Isomorphism_is_Isomorphism_in_Category_CLat | https://proofwiki.org/wiki/Complete_Lattice_Isomorphism_is_Isomorphism_in_Category_CLat | [
"Complete Lattice Isomorphisms",
"Category of Complete Lattices"
] | [
"Definition:Category of Complete Lattices",
"Definition:Morphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Complete Lattice Isomorphism"
] | [
"Definition:Complete Lattice",
"Definition:Category of Complete Lattices",
"Definition:Complete Lattice Homomorphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)",
"Definition:Morphism",
"Definition:Identity Morphism",
"Definition:Category of Complete Lattice... |
proofwiki-22617 | Transition Mapping between Charts is Homeomorphism | Let $M$ be a topological space.
Let $d$ be a natural number.
Let $\struct {U, \phi}$ and $\struct {V, \psi}$ be $d$-dimensional charts of $M$
such that $U \cap V \ne \O$.
Then, the transition map from $\phi$ to $\psi$:
:$\psi \circ \phi^{-1} : \map \phi {U \cap V} \to \map \psi {U \cap V}$
is an homeomorphism. | By definition of charts, $U$ and $V$ are homeomorphisms.
By Inverse of Homeomorphism is Homeomorphism, $\phi^{-1}$ is an homeomorphism.
By Composite of Homeomorphisms is Homeomorphism, $\psi \circ \phi^{-1}$ is an homeomorphism.
{{qed}}
Category:Homeomorphisms (Topological Spaces)
p8fndu1hk29p5ei5ro2n490ornmjegf | Let $M$ be a [[Definition:Topological Space|topological space]].
Let $d$ be a [[Definition:Natural Number|natural number]].
Let $\struct {U, \phi}$ and $\struct {V, \psi}$ be $d$-[[Definition:Dimension of Chart|dimensional]] [[Definition:Chart|charts]] of $M$
such that $U \cap V \ne \O$.
Then, the [[Definition:Tra... | By definition of [[Definition:Chart|charts]], $U$ and $V$ are [[Definition:Homeomorphism|homeomorphism]]s.
By [[Inverse of Homeomorphism is Homeomorphism]], $\phi^{-1}$ is an [[Definition:Homeomorphism|homeomorphism]].
By [[Composite of Homeomorphisms is Homeomorphism]], $\psi \circ \phi^{-1}$ is an [[Definition:Home... | Transition Mapping between Charts is Homeomorphism | https://proofwiki.org/wiki/Transition_Mapping_between_Charts_is_Homeomorphism | https://proofwiki.org/wiki/Transition_Mapping_between_Charts_is_Homeomorphism | [
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Natural Numbers",
"Definition:Chart",
"Definition:Chart",
"Definition:Transition Mapping between Charts",
"Definition:Homeomorphism"
] | [
"Definition:Chart",
"Definition:Homeomorphism",
"Inverse of Homeomorphism is Homeomorphism",
"Definition:Homeomorphism",
"Composite of Homeomorphisms is Homeomorphism",
"Definition:Homeomorphism",
"Category:Homeomorphisms (Topological Spaces)"
] |
proofwiki-22618 | Algebraic Numbers form Countable Subfield of Complex Numbers | The set $\Bbb A$ of algebraic numbers forms a countable subfield of the field of complex numbers. | By definition, $\Bbb A$ is the subset of the complex numbers which consists of roots of polynomials with coefficients in $\Q$.
From Algebraic Numbers form Field, $\Bbb A$ is a field.
The result follows by definition of subfield.
{{qed}} | The [[Definition:Set|set]] $\Bbb A$ of [[Definition:Algebraic Number|algebraic numbers]] forms a [[Definition:Countable Set|countable]] [[Definition:Subfield|subfield]] of the [[Definition:Field of Complex Numbers|field of complex numbers]]. | By definition, $\Bbb A$ is the [[Definition:Subset|subset]] of the [[Definition:Complex Number|complex numbers]] which consists of [[Definition:Root of Polynomial|roots of polynomials]] with coefficients in $\Q$.
From [[Algebraic Numbers form Field]], $\Bbb A$ is a [[Definition:Field (Abstract Algebra)|field]].
The ... | Algebraic Numbers form Countable Subfield of Complex Numbers | https://proofwiki.org/wiki/Algebraic_Numbers_form_Countable_Subfield_of_Complex_Numbers | https://proofwiki.org/wiki/Algebraic_Numbers_form_Countable_Subfield_of_Complex_Numbers | [
"Algebraic Numbers",
"Complex Numbers",
"Countable Sets",
"Examples of Subfields"
] | [
"Definition:Set",
"Definition:Algebraic Number",
"Definition:Countable Set",
"Definition:Subfield",
"Definition:Field of Complex Numbers"
] | [
"Definition:Subset",
"Definition:Complex Number",
"Definition:Root of Polynomial",
"Algebraic Numbers form Field",
"Definition:Field (Abstract Algebra)",
"Definition:Subfield"
] |
proofwiki-22619 | Complete Lattice is Lattice | Let $L= \struct{S, \preceq}$ be a complete lattice.
Then:
:$\struct {S, \vee, \wedge, \preceq}$ is a lattice
where $\vee$ and $\wedge$ denote the join and meet operations on $S$, respectively. | By definition of complete lattice:
:$\forall T \subseteq S: T$ admits both a supremum and an infimum in $L$
Hence:
:$\forall x, y \in S$, the join and meet of $x$ and $y$ exist in $L$
Denote with $\vee$ and $\wedge$ the join and meet operations on $S$, respectively.
It follows that $\struct {S, \vee, \wedge, \preceq}$ ... | Let $L= \struct{S, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Then:
:$\struct {S, \vee, \wedge, \preceq}$ is a [[Definition:Lattice (Order Theory)|lattice]]
where $\vee$ and $\wedge$ denote the [[Definition:Join (Order Theory)|join]] and [[Definition:Meet (Order Theory)|meet]] operations on $S$,... | By definition of [[Definition:Complete Lattice|complete lattice]]:
:$\forall T \subseteq S: T$ admits both a [[Definition:Supremum of Set|supremum]] and an [[Definition:Infimum of Set|infimum]] in $L$
Hence:
:$\forall x, y \in S$, the [[Definition:Join (Order Theory)|join]] and [[Definition:Meet (Order Theory)|meet]]... | Complete Lattice is Lattice | https://proofwiki.org/wiki/Complete_Lattice_is_Lattice | https://proofwiki.org/wiki/Complete_Lattice_is_Lattice | [
"Complete Lattices",
"Lattices (Order Theory)"
] | [
"Definition:Complete Lattice",
"Definition:Lattice (Order Theory)",
"Definition:Join (Order Theory)",
"Definition:Meet (Order Theory)"
] | [
"Definition:Complete Lattice",
"Definition:Supremum of Set",
"Definition:Infimum of Set",
"Definition:Join (Order Theory)",
"Definition:Meet (Order Theory)",
"Definition:Join (Order Theory)",
"Definition:Meet (Order Theory)",
"Definition:Lattice (Order Theory)",
"Category:Complete Lattices",
"Cate... |
proofwiki-22620 | Complete Lattice Homomorphism is Frame Homomorphism | Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be complete lattices.
Let $\phi: L_1 \to L_2$ be a complete lattice homomorphsim between $L_1$ and $L_2$.
Then:
:$\phi$ is a frame homomorphism | By definition of complete lattice homomorphsim:
:$\phi$ satisfies the complete lattice homomorphism axioms:
{{:Axiom:Complete Lattice Homomorphism Axioms}}
It follows that $\phi$ is finite meet preserving:
:$\forall$ finite $A \subseteq S_1 : \map \phi {\inf A} = \inf \set{\map \phi a : a \in A}$
Hence $\phi$ is finite... | Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be [[Definition:Complete Lattice|complete lattices]].
Let $\phi: L_1 \to L_2$ be a [[Definition:Complete Lattice Homomorphism|complete lattice homomorphsim]] between $L_1$ and $L_2$.
Then:
:$\phi$ is a [[Definition:Frame Homomorphism|frame hom... | By definition of [[Definition:Complete Lattice Homomorphism|complete lattice homomorphsim]]:
:$\phi$ satisfies the [[Axiom:Complete Lattice Homomorphism Axioms|complete lattice homomorphism axioms]]:
{{:Axiom:Complete Lattice Homomorphism Axioms}}
It follows that $\phi$ is [[Definition:Finite Meet Preserving Mapping|... | Complete Lattice Homomorphism is Frame Homomorphism | https://proofwiki.org/wiki/Complete_Lattice_Homomorphism_is_Frame_Homomorphism | https://proofwiki.org/wiki/Complete_Lattice_Homomorphism_is_Frame_Homomorphism | [
"Complete Lattice Homomorphisms",
"Frame Homomorphisms"
] | [
"Definition:Complete Lattice",
"Definition:Complete Lattice Homomorphism",
"Definition:Frame Homomorphism"
] | [
"Definition:Complete Lattice Homomorphism",
"Axiom:Complete Lattice Homomorphism Axioms",
"Definition:Finite Meet Preserving Mapping",
"Definition:Finite Set",
"Definition:Finite Meet Preserving Mapping",
"Definition:Arbitrary Join Preserving Mapping",
"Definition:Frame Homomorphism",
"Category:Comple... |
proofwiki-22621 | Algorithmic Complexity of Euclidean Algorithm | Let $a, b \in \Z_{>0}$ be (strictly) positive integers.
Let the Euclidean Algorithm be employed to find the GCD of $a$ and $b$.
Let $\CC$ be the algorithmic complexity of this operation.
Then:
:$\CC = \map \OO {\map \ln {\max \set {a, b} } }$
where $\OO$ denotes big-$\OO$ notation. | Lamé's Theorem states that:
{{:Lamé's Theorem}}
It follows that:
{{begin-eqn}}
{{eqn | l = \CC
| r = \map \OO {\map \log { \min \set {a, b} } }
}}
{{eqn | r = \map \OO {\map \log { \max \set {a, b} } }
| c = $\min \set {a, b} \leq \max \set {a, b}$
}}
{{eqn | r = \map \OO {\log e \map \ln { \max \set {a, b}... | Let $a, b \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let the [[Euclidean Algorithm]] be employed to find the [[Definition:GCD of Integers|GCD]] of $a$ and $b$.
Let $\CC$ be the [[Definition:Algorithmic Complexity|algorithmic complexity]] of this operation.
Then:
:$\CC = \... | [[Lamé's Theorem]] states that:
{{:Lamé's Theorem}}
It follows that:
{{begin-eqn}}
{{eqn | l = \CC
| r = \map \OO {\map \log { \min \set {a, b} } }
}}
{{eqn | r = \map \OO {\map \log { \max \set {a, b} } }
| c = $\min \set {a, b} \leq \max \set {a, b}$
}}
{{eqn | r = \map \OO {\log e \map \ln { \max \set... | Algorithmic Complexity of Euclidean Algorithm | https://proofwiki.org/wiki/Algorithmic_Complexity_of_Euclidean_Algorithm | https://proofwiki.org/wiki/Algorithmic_Complexity_of_Euclidean_Algorithm | [
"Euclidean Algorithm",
"Algorithmic Complexity"
] | [
"Definition:Strictly Positive/Integer",
"Euclidean Algorithm",
"Definition:Greatest Common Divisor/Integers",
"Definition:Algorithm/Analysis/Complexity",
"Definition:Big-O Notation"
] | [
"Lamé's Theorem",
"Change of Base of Logarithm",
"Definition:Constant"
] |
proofwiki-22622 | Analytic Real Function has Derivatives of All Orders | Let $f$ be a real function which is analytic real function on the open interval $\openint a b$.
Then $f$ has continuous derivatives of all orders. | {{Recall|Analytic Real Function}}
{{:Definition:Analytic Real Function}}
Hence, in order for $f$ to be analytic, $f$ is {{afortiori}} smooth.
Hence, by definition of smooth, $f$ has continuous derivatives of all orders.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Analytic Real Function|analytic real function]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Then $f$ has [[Definition:Continuous Real Function|continuous]] [[Definition:Derivative|derivatives]] of all [[Definition:... | {{Recall|Analytic Real Function}}
{{:Definition:Analytic Real Function}}
Hence, in order for $f$ to be [[Definition:Analytic Real Function|analytic]], $f$ is {{afortiori}} [[Definition:Smooth Real Function|smooth]].
Hence, by definition of [[Definition:Smooth Real Function|smooth]], $f$ has [[Definition:Continuous Re... | Analytic Real Function has Derivatives of All Orders | https://proofwiki.org/wiki/Analytic_Real_Function_has_Derivatives_of_All_Orders | https://proofwiki.org/wiki/Analytic_Real_Function_has_Derivatives_of_All_Orders | [
"Analytic Real Functions",
"Smooth Real Functions"
] | [
"Definition:Real Function",
"Definition:Analytic Function/Real Numbers",
"Definition:Real Interval/Open",
"Definition:Continuous Real Function",
"Definition:Derivative",
"Definition:Derivative/Higher Derivatives/Order of Derivative"
] | [
"Definition:Analytic Function/Real Numbers",
"Definition:Smooth Real Function",
"Definition:Smooth Real Function",
"Definition:Continuous Real Function",
"Definition:Derivative",
"Definition:Derivative/Higher Derivatives/Order of Derivative"
] |
proofwiki-22623 | Smooth Real Function is not necessarily Analytic | Let $f$ be a real function which is smooth on the open interval $\openint a b$.
real function which is analytic real function on the open interval $\openint a b$.
Then it is not necessarily the case that $f$ is also analytic on $\openint a b$. | ;Proof by Counterexample
Consider the real function $f: \R \to \R$ defined as:
:$\forall x \in \R: \map f x = \begin {cases} \map \exp {\dfrac {-1} {x^2} } & : x \ne 0 \\ 0 & : x = 0 \end {cases}$
$f$ is of differentiability class $C^\infty$ such that the derivatives of all orders equal $0$.
Thus the Taylor series of $... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Smooth Real Function|smooth]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
[[Definition:Real Function|real function]] which is [[Definition:Analytic Real Function|analytic real function]] on the [[Definition:Open Re... | ;[[Proof by Counterexample]]
Consider the [[Definition:Real Function|real function]] $f: \R \to \R$ defined as:
:$\forall x \in \R: \map f x = \begin {cases} \map \exp {\dfrac {-1} {x^2} } & : x \ne 0 \\ 0 & : x = 0 \end {cases}$
$f$ is of [[Definition:Differentiability Class|differentiability class]] $C^\infty$ such... | Smooth Real Function is not necessarily Analytic | https://proofwiki.org/wiki/Smooth_Real_Function_is_not_necessarily_Analytic | https://proofwiki.org/wiki/Smooth_Real_Function_is_not_necessarily_Analytic | [
"Analytic Real Functions",
"Smooth Real Functions"
] | [
"Definition:Real Function",
"Definition:Smooth Real Function",
"Definition:Real Interval/Open",
"Definition:Real Function",
"Definition:Analytic Function/Real Numbers",
"Definition:Real Interval/Open",
"Definition:Analytic Function/Real Numbers"
] | [
"Proof by Counterexample",
"Definition:Real Function",
"Definition:Differentiability Class",
"Definition:Derivative",
"Definition:Derivative/Higher Derivatives/Order of Derivative",
"Definition:Taylor Series",
"Definition:Convergent Series/Number Field",
"Definition:Analytic Function/Real Numbers"
] |
proofwiki-22624 | Conditionally Convergent Series has Infinitely Many Positive and Negative Terms | Let $S = \sequence {s_n}$ be a real infinite series which is conditionally convergent.
Then it has infinitely many strictly positive and negative terms. | {{WLOG}}, we show there are infinitely many negative terms.
{{AimForCont}} there are finitely many negative terms.
Let $N = \sequence {n_k}$ be the sequence defined as:
:$n_k = \sequence {\size {s_k} }$
where $\size {s_k}$ denotes the absolute value of $s_k$.
By Convergent Sequence with Finite Number of Terms Deleted i... | Let $S = \sequence {s_n}$ be a [[Definition:Real Series|real]] [[Definition:Infinite Series|infinite series]] which is [[Definition:Conditionally Convergent Series|conditionally convergent]].
Then it has [[Definition:Infinite Set|infinitely many]] [[Definition:Strictly Positive Real Number|strictly positive]] and [[D... | {{WLOG}}, we show there are [[Definition:Infinite Set|infinitely many]] [[Definition:Negative Real Number|negative]] [[Definition:Term of Sequence|terms]].
{{AimForCont}} there are [[Definition:Finite Set|finitely many]] [[Definition:Negative Real Number|negative]] [[Definition:Term of Sequence|terms]].
Let $N = \seq... | Conditionally Convergent Series has Infinitely Many Positive and Negative Terms | https://proofwiki.org/wiki/Conditionally_Convergent_Series_has_Infinitely_Many_Positive_and_Negative_Terms | https://proofwiki.org/wiki/Conditionally_Convergent_Series_has_Infinitely_Many_Positive_and_Negative_Terms | [
"Convergent Series"
] | [
"Definition:Series/Real",
"Definition:Series",
"Definition:Conditionally Convergent Series",
"Definition:Infinite Set",
"Definition:Strictly Positive/Real Number",
"Definition:Negative/Real Number",
"Definition:Term of Sequence"
] | [
"Definition:Infinite Set",
"Definition:Negative/Real Number",
"Definition:Term of Sequence",
"Definition:Finite Set",
"Definition:Negative/Real Number",
"Definition:Term of Sequence",
"Definition:Real Sequence",
"Definition:Absolute Value",
"Convergent Sequence with Finite Number of Terms Deleted is... |
proofwiki-22625 | Sum of Logarithms/Complex Logarithm | Let $x, y \in \C$ where $x = r_1 e^{i \theta_1}$ and $y = r_2 e^{i \theta_2}$
Where:
:$r_1$ and $r_2$ are both (strictly) positive real numbers.
Then:
:$\ln x + \ln y = \map \ln {x y}$
where $\ln$ denotes the complex natural logarithm. | We have:
{{begin-eqn}}
{{eqn | l = x
| r = r_1 e^{i \theta_1}
| c =
}}
{{eqn | ll = \leadsto
| l = \ln x
| r = \map \ln {r_1} + i \paren {\theta_1 + 2 k \pi}
| c = {{Defof|Complex Natural Logarithm}}
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = y
| r = r_2 e^{i \theta_2}
| c ... | Let $x, y \in \C$ where $x = r_1 e^{i \theta_1}$ and $y = r_2 e^{i \theta_2}$
Where:
:$r_1$ and $r_2$ are both [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]].
Then:
:$\ln x + \ln y = \map \ln {x y}$
where $\ln$ denotes the [[Definition:Complex Natural Logarithm|complex natural logarithm... | We have:
{{begin-eqn}}
{{eqn | l = x
| r = r_1 e^{i \theta_1}
| c =
}}
{{eqn | ll = \leadsto
| l = \ln x
| r = \map \ln {r_1} + i \paren {\theta_1 + 2 k \pi}
| c = {{Defof|Complex Natural Logarithm}}
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = y
| r = r_2 e^{i \theta_2}
| c... | Sum of Logarithms/Complex Logarithm | https://proofwiki.org/wiki/Sum_of_Logarithms/Complex_Logarithm | https://proofwiki.org/wiki/Sum_of_Logarithms/Complex_Logarithm | [
"Sum of Logarithms"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Natural Logarithm/Complex"
] | [
"Exponent Combination Laws/Product of Powers",
"Sum of Logarithms/Natural Logarithm",
"Category:Sum of Logarithms"
] |
proofwiki-22626 | Hall's Theorem (Group Theory) | Let $G$ be a solvable group of order $ab$ such that $a$ and $b$ are coprime.
Then:
: $G$ contains a subgroup of order $a$,
and
: subgroups of order $a$ are mutually conjugate. | {{ProofWanted}}
{{Namedfor|Philip Hall|cat = Hall}} | Let $G$ be a [[Definition:Solvable Group|solvable group]] of [[Definition:Order of Structure|order]] $ab$ such that $a$ and $b$ are [[Definition:Coprime Integers|coprime]].
Then:
: $G$ contains a [[Definition:Subgroup|subgroup]] of [[Definition:Order of Group Element|order]] $a$,
and
: [[Definition:Subgroup|subgroups... | {{ProofWanted}}
{{Namedfor|Philip Hall|cat = Hall}} | Hall's Theorem (Group Theory) | https://proofwiki.org/wiki/Hall's_Theorem_(Group_Theory) | https://proofwiki.org/wiki/Hall's_Theorem_(Group_Theory) | [
"Solvable Groups"
] | [
"Definition:Solvable Group",
"Definition:Order of Structure",
"Definition:Coprime/Integers",
"Definition:Subgroup",
"Definition:Order of Group Element",
"Definition:Subgroup",
"Definition:Order of Group Element",
"Conjugate of Subgroup is Subgroup"
] | [] |
proofwiki-22627 | Antilogarithm Function is Exponential Function | Let $y = \operatorname {alog}_b x$ be the antilogarithm of $x$ base $b$.
Then:
:$y = b^x$ | {{begin-eqn}}
{{eqn | l = y
| r = \operatorname {alog}_b x
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = x
| r = \log_b y
| c = {{Defof|Antilogarithm}}
}}
{{eqn | ll= \leadsto
| l = b^x
| r = y
| c = {{Defof|General Logarithm}}
}}
{{end-eqn}}
{{qed}} | Let $y = \operatorname {alog}_b x$ be the [[Definition:Antilogarithm|antilogarithm of $x$ base $b$]].
Then:
:$y = b^x$ | {{begin-eqn}}
{{eqn | l = y
| r = \operatorname {alog}_b x
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = x
| r = \log_b y
| c = {{Defof|Antilogarithm}}
}}
{{eqn | ll= \leadsto
| l = b^x
| r = y
| c = {{Defof|General Logarithm}}
}}
{{end-eqn}}
{{qed}} | Antilogarithm Function is Exponential Function | https://proofwiki.org/wiki/Antilogarithm_Function_is_Exponential_Function | https://proofwiki.org/wiki/Antilogarithm_Function_is_Exponential_Function | [
"Antilogarithms",
"Exponential Function"
] | [
"Definition:Antilogarithm"
] | [] |
proofwiki-22628 | Composite Complete Lattice Homomorphisms is Complete Lattice Homomorphism | Let $L_1 = \struct{A_1, \preceq_1}$, $L_2 = \struct{A_2, \preceq_2}$ and $L_3 = \struct{A_3, \preceq_3}$ be complete lattices.
Let $\phi_1: L_1 \to L_2$ and $\phi_2: L_2 \to L_3$ be complete lattice homomorphisms.
Let $\phi_2 \circ \phi_1 : A_1 \to A_3$ be the composite mapping of $\phi_1$ and $\phi_2$
Then:
:$\phi_2... | === $\phi_2 \circ \phi_1$ is arbitrary join preserving ===
We have:
{{begin-eqn}}
{{eqn | q = \forall A \subseteq A_1
| l = \map {\phi_2 \circ \phi_1} {\sup A}
| r = \map {\phi_2} {\map {\phi_1} {\sup A} }
| c = {{Defof|Composite Mapping}}
}}
{{eqn | r = \map {\phi_2} {\sup \paren{\phi_1 \sqbrk A} }
... | Let $L_1 = \struct{A_1, \preceq_1}$, $L_2 = \struct{A_2, \preceq_2}$ and $L_3 = \struct{A_3, \preceq_3}$ be [[Definition:Complete Lattice|complete lattices]].
Let $\phi_1: L_1 \to L_2$ and $\phi_2: L_2 \to L_3$ be [[Definition:Complete Lattice Homomorphism|complete lattice homomorphisms]].
Let $\phi_2 \circ \phi_1 :... | === $\phi_2 \circ \phi_1$ is arbitrary join preserving ===
We have:
{{begin-eqn}}
{{eqn | q = \forall A \subseteq A_1
| l = \map {\phi_2 \circ \phi_1} {\sup A}
| r = \map {\phi_2} {\map {\phi_1} {\sup A} }
| c = {{Defof|Composite Mapping}}
}}
{{eqn | r = \map {\phi_2} {\sup \paren{\phi_1 \sqbrk A} }
... | Composite Complete Lattice Homomorphisms is Complete Lattice Homomorphism | https://proofwiki.org/wiki/Composite_Complete_Lattice_Homomorphisms_is_Complete_Lattice_Homomorphism | https://proofwiki.org/wiki/Composite_Complete_Lattice_Homomorphisms_is_Complete_Lattice_Homomorphism | [
"Complete Lattice Homomorphisms"
] | [
"Definition:Complete Lattice",
"Definition:Complete Lattice Homomorphism",
"Definition:Composition of Mappings",
"Definition:Complete Lattice Homomorphism"
] | [
"Image of Subset under Composite Relation with Common Codomain and Domain",
"Definition:Arbitrary Join Preserving Mapping",
"Definition:Arbitrary Join Preserving Mapping"
] |
proofwiki-22629 | Identity Mapping is Complete Lattice Homomorphism | Let $L = \struct{A, \preceq}$ be a complete lattice.
Let $\operatorname{id}_A$ denote the identity mapping on $A$.
Then:
:$\operatorname{id}_A$ is a complete lattice homomorphism of $L$ to $L$ | === $\operatorname{id}_A$ is arbitrary join preserving ===
We have:
{{begin-eqn}}
{{eqn | q = \forall X \subseteq A
| l = \map {\operatorname{id}_A} {\sup X}
| r = \sup \set{x : x \in X}
}}
{{eqn | r = \sup \set{\map{\operatorname{id}_A} x : x \in X}
}}
{{end-eqn}}
It follows that $\operatorname{id}_A$ is a... | Let $L = \struct{A, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $\operatorname{id}_A$ denote the [[Definition:Identity Mapping|identity mapping]] on $A$.
Then:
:$\operatorname{id}_A$ is a [[Definition:Complete Lattice Homomorphism|complete lattice homomorphism]] of $L$ to $L$ | === $\operatorname{id}_A$ is arbitrary join preserving ===
We have:
{{begin-eqn}}
{{eqn | q = \forall X \subseteq A
| l = \map {\operatorname{id}_A} {\sup X}
| r = \sup \set{x : x \in X}
}}
{{eqn | r = \sup \set{\map{\operatorname{id}_A} x : x \in X}
}}
{{end-eqn}}
It follows that $\operatorname{id}_A$ is... | Identity Mapping is Complete Lattice Homomorphism | https://proofwiki.org/wiki/Identity_Mapping_is_Complete_Lattice_Homomorphism | https://proofwiki.org/wiki/Identity_Mapping_is_Complete_Lattice_Homomorphism | [
"Complete Lattice Homomorphisms"
] | [
"Definition:Complete Lattice",
"Definition:Identity Mapping",
"Definition:Complete Lattice Homomorphism"
] | [
"Definition:Arbitrary Join Preserving Mapping",
"Definition:Arbitrary Join Preserving Mapping"
] |
proofwiki-22630 | Uniform Prism is Isogonal | Let $P$ be a uniform prism.
Then $P$ is isogonal. | {{Recall|Uniform Prism}}
{{:Definition:Uniform Prism}}
Hence $P$ is a regular prism {{afortiori}}.
From Regular Prism is Isogonal, $P$ is isogonal.
Hence the result.
{{qed}}
Category:Uniform Prisms
Category:Isogonal Polyhedra
ki33ojoet4fycuuk5eih4xxctx3doz3 | Let $P$ be a [[Definition:Uniform Prism|uniform prism]].
Then $P$ is [[Definition:Isogonal Polyhedron|isogonal]]. | {{Recall|Uniform Prism}}
{{:Definition:Uniform Prism}}
Hence $P$ is a [[Definition:Regular Prism|regular prism]] {{afortiori}}.
From [[Regular Prism is Isogonal]], $P$ is [[Definition:Isogonal Polyhedron|isogonal]].
Hence the result.
{{qed}}
[[Category:Uniform Prisms]]
[[Category:Isogonal Polyhedra]]
ki33ojoet4fycu... | Uniform Prism is Isogonal | https://proofwiki.org/wiki/Uniform_Prism_is_Isogonal | https://proofwiki.org/wiki/Uniform_Prism_is_Isogonal | [
"Uniform Prisms",
"Isogonal Polyhedra",
"Uniform Prisms",
"Isogonal Polyhedra"
] | [
"Definition:Uniform Prism",
"Definition:Isogonal Polyhedron"
] | [
"Definition:Regular Prism",
"Regular Prism is Isogonal",
"Definition:Isogonal Polyhedron",
"Category:Uniform Prisms",
"Category:Isogonal Polyhedra"
] |
proofwiki-22631 | Vertical Composition of Natural Transformations is Natural Transformation | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F, G, H : \mathbf C \to \mathbf D$ be covariant functors.
Let $\eta: F \to G$ and $\xi: G \to H$ be natural transformations.
Then, the vertical composition of $\eta$ and $\xi$:
:$\xi \circ \eta : F \to H$
with
:$\paren {\xi \circ \eta}_X = \xi_X \circ \eta_X$ for each... | By definiition of natural transformation and vertical composition:
:For each morphism $f : X \to Y$ in $\mathbf C$, the diagram:
::<nowiki>$\xymatrix{
F X \ar[d]^{\eta_X} \ar@/_2em/[dd]_{\paren{\xi \circ \eta}_X} \ar[r]^{F f} & F Y \ar[d]^{\eta_Y} \ar@/^2em/[dd]^{\paren{\xi \circ \eta}_Y} \\
G X \ar[d]^{\xi_X} \ar... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F, G, H : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta: F \to G$ and $\xi: G \to H$ be [[Definition:Natural Transformation|natural transformations]].
Then, the [[Definition:Vertical Composition ... | By definiition of [[Definition:Natural Transformation|natural transformation]] and [[Definition:Vertical Composition of Natural Transformations|vertical composition]]:
:For each [[Definition:Morphism (Category Theory)|morphism]] $f : X \to Y$ in $\mathbf C$, the diagram:
::<nowiki>$\xymatrix{
F X \ar[d]^{\eta_X} \ar... | Vertical Composition of Natural Transformations is Natural Transformation/Proof 1 | https://proofwiki.org/wiki/Vertical_Composition_of_Natural_Transformations_is_Natural_Transformation | https://proofwiki.org/wiki/Vertical_Composition_of_Natural_Transformations_is_Natural_Transformation/Proof_1 | [
"Vertical Composition of Natural Transformations is Natural Transformation",
"Natural Transformations"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Vertical Composition of Natural Transformations",
"Definition:Object (Category Theory)",
"Definition:Natural Transformation"
] | [
"Definition:Natural Transformation",
"Definition:Vertical Composition of Natural Transformations",
"Definition:Morphism",
"Definition:Commutative Diagram",
"Definition:Morphism",
"Definition:Vertical Composition of Natural Transformations",
"Definition:Natural Transformation"
] |
proofwiki-22632 | Vertical Composition of Natural Transformations is Natural Transformation | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F, G, H : \mathbf C \to \mathbf D$ be covariant functors.
Let $\eta: F \to G$ and $\xi: G \to H$ be natural transformations.
Then, the vertical composition of $\eta$ and $\xi$:
:$\xi \circ \eta : F \to H$
with
:$\paren {\xi \circ \eta}_X = \xi_X \circ \eta_X$ for each... | For each morphism $f : X \to Y$ in $\mathbf C$, we have:
{{begin-eqn}}
{{eqn | l = H f \circ \paren{\xi \circ \eta}_X
| r = H f \circ \paren{\xi_X \circ \eta_X}
| c = {{Defof|Vertical Composition of Natural Transformations}}
}}
{{eqn | r = \paren{H f \circ \xi_X} \circ \eta_X
| c = Associtivity of Com... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F, G, H : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta: F \to G$ and $\xi: G \to H$ be [[Definition:Natural Transformation|natural transformations]].
Then, the [[Definition:Vertical Composition ... | For each [[Definition:Morphism (Category Theory)|morphism]] $f : X \to Y$ in $\mathbf C$, we have:
{{begin-eqn}}
{{eqn | l = H f \circ \paren{\xi \circ \eta}_X
| r = H f \circ \paren{\xi_X \circ \eta_X}
| c = {{Defof|Vertical Composition of Natural Transformations}}
}}
{{eqn | r = \paren{H f \circ \xi_X} \c... | Vertical Composition of Natural Transformations is Natural Transformation/Proof 2 | https://proofwiki.org/wiki/Vertical_Composition_of_Natural_Transformations_is_Natural_Transformation | https://proofwiki.org/wiki/Vertical_Composition_of_Natural_Transformations_is_Natural_Transformation/Proof_2 | [
"Vertical Composition of Natural Transformations is Natural Transformation",
"Natural Transformations"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Vertical Composition of Natural Transformations",
"Definition:Object (Category Theory)",
"Definition:Natural Transformation"
] | [
"Definition:Morphism",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Vertical Composition of Natural Transformations",
"Definition:Natural Transformation"
] |
proofwiki-22633 | Uniform Antiprism is Isogonal | Let $P$ be a uniform antiprism.
Then $P$ is isogonal. | {{Recall|Uniform Antiprism}}
{{:Definition:Uniform Antiprism}}
An equilateral triangle is a special case of an isosceles triangle.
{{Recall|Right-Regular Antiprism}}
{{:Definition:Right-Regular Antiprism}}
Hence $P$ is a right-regular antiprism {{afortiori}}.
From Right-Regular Antiprism is Isogonal, $P$ is isogonal.
H... | Let $P$ be a [[Definition:Uniform Antiprism|uniform antiprism]].
Then $P$ is [[Definition:Isogonal Polyhedron|isogonal]]. | {{Recall|Uniform Antiprism}}
{{:Definition:Uniform Antiprism}}
An [[Definition:Equilateral Triangle|equilateral triangle]] is a special case of an [[Definition:Isosceles Triangle|isosceles triangle]].
{{Recall|Right-Regular Antiprism}}
{{:Definition:Right-Regular Antiprism}}
Hence $P$ is a [[Definition:Right-Regula... | Uniform Antiprism is Isogonal | https://proofwiki.org/wiki/Uniform_Antiprism_is_Isogonal | https://proofwiki.org/wiki/Uniform_Antiprism_is_Isogonal | [
"Uniform Antiprisms",
"Isogonal Polyhedra"
] | [
"Definition:Uniform Antiprism",
"Definition:Isogonal Polyhedron"
] | [
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Right-Regular Antiprism",
"Right-Regular Antiprism is Isogonal",
"Definition:Isogonal Polyhedron",
"Category:Uniform Antiprisms",
"Category:Isogonal Polyhedra"
] |
proofwiki-22634 | Grothendieck Universe is Closed under Finite Union | Let $\mathbb U$ be a Grothendieck universe.
Let $\set {u_1, u_2, \cdots, u_n : n \in \N} \in \mathbb U$ be a finite subset of elements of $\mathbb U$
Then $\ds \bigcup_{i \mathop \in \N} u_i \in \mathbb U$ | If $\mathbb U = \O$, the claim is true vacuously.
Assume $\mathbb U \ne \O$.
By Nonempty Grothendieck Universe contains Von Neumann Natural Numbers, $\N \subseteq \mathbb U$.
Grothendieck Universe: Axiom $(4)$ is:
:If $A \in \mathbb U$, and $\set {u_\alpha: \alpha \in A}$ is a family of elements $u_\alpha \in \mathbb U... | Let $\mathbb U$ be a [[Definition:Grothendieck Universe|Grothendieck universe]].
Let $\set {u_1, u_2, \cdots, u_n : n \in \N} \in \mathbb U$ be a [[Definition:Finite Subset|finite subset]] of [[Definition:Element|elements]] of $\mathbb U$
Then $\ds \bigcup_{i \mathop \in \N} u_i \in \mathbb U$ | If $\mathbb U = \O$, the claim is true [[Definition:Vacuous Truth|vacuously]].
Assume $\mathbb U \ne \O$.
By [[Nonempty Grothendieck Universe contains Von Neumann Natural Numbers]], $\N \subseteq \mathbb U$.
[[Definition:Grothendieck Universe|Grothendieck Universe: Axiom $(4)$]] is:
:If $A \in \mathbb U$, and $\se... | Grothendieck Universe is Closed under Finite Union | https://proofwiki.org/wiki/Grothendieck_Universe_is_Closed_under_Finite_Union | https://proofwiki.org/wiki/Grothendieck_Universe_is_Closed_under_Finite_Union | [
"Grothendieck Universes"
] | [
"Definition:Grothendieck Universe",
"Definition:Finite Subset",
"Definition:Element"
] | [
"Definition:Vacuous Truth",
"Nonempty Grothendieck Universe contains Von Neumann Natural Numbers",
"Definition:Grothendieck Universe",
"Definition:Indexing Set/Family of Sets",
"Definition:Element",
"Category:Grothendieck Universes"
] |
proofwiki-22635 | Arrow's Impossibility Theorem | Let voters be given $3$ or more distinct choices to choose from.
Then there is no ranking system which can aggregate the individual preferences of two or more individuals so that $4$ apparently reasonable conditions are met. | {{ProofWanted}}
{{Namedfor|Kenneth Joseph Arrow|cat = Arrow}} | Let [[Definition:Voter|voters]] be given $3$ or more distinct [[Definition:Choice|choices]] to choose from.
Then there is no [[Definition:Ranking|ranking]] system which can aggregate the individual preferences of two or more individuals so that $4$ apparently reasonable conditions are met. | {{ProofWanted}}
{{Namedfor|Kenneth Joseph Arrow|cat = Arrow}} | Arrow's Impossibility Theorem | https://proofwiki.org/wiki/Arrow's_Impossibility_Theorem | https://proofwiki.org/wiki/Arrow's_Impossibility_Theorem | [
"Game Theory"
] | [
"Definition:Voter",
"Definition:Choice",
"Definition:Ranking"
] | [] |
proofwiki-22636 | Frame is Distributive Lattice | Let $L= \struct{S, \preceq}$ be a frame.
Then:
:$\struct {S, \vee, \wedge, \preceq}$ is a distributive lattice
where $\vee$ and $\wedge$ denote the join and meet operations on $S$, respectively. | By definition of frame:
:$L$ is a complete lattice
From Complete Lattice is Lattice:
:$\struct {S, \vee, \wedge, \preceq}$ is a lattice
By definition of frame:
{{begin-axiom}}
{{axiom | q = \forall a \in S, B \subseteq S
| m = a \wedge \bigvee B = \bigvee \set {a \wedge b : b \in B}
}}
{{end-axiom}}
Hence:
{{be... | Let $L= \struct{S, \preceq}$ be a [[Definition:Frame (Lattice Theory)|frame]].
Then:
:$\struct {S, \vee, \wedge, \preceq}$ is a [[Definition:Distributive Lattice|distributive lattice]]
where $\vee$ and $\wedge$ denote the [[Definition:Join (Order Theory)|join]] and [[Definition:Meet (Order Theory)|meet]] operations o... | By definition of [[Definition:Frame (Lattice Theory)|frame]]:
:$L$ is a [[Definition:Complete Lattice|complete lattice]]
From [[Complete Lattice is Lattice]]:
:$\struct {S, \vee, \wedge, \preceq}$ is a [[Definition:Lattice (Order Theory)|lattice]]
By definition of [[Definition:Frame (Lattice Theory)|frame]]:
{{begin... | Frame is Distributive Lattice | https://proofwiki.org/wiki/Frame_is_Distributive_Lattice | https://proofwiki.org/wiki/Frame_is_Distributive_Lattice | [
"Frames",
"Distributive Lattices"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Distributive Lattice",
"Definition:Join (Order Theory)",
"Definition:Meet (Order Theory)"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Complete Lattice",
"Complete Lattice is Lattice",
"Definition:Lattice (Order Theory)",
"Definition:Frame (Lattice Theory)",
"Definition:Distributive Lattice",
"Definition:Distributive Lattice",
"Category:Frames",
"Category:Distributive Lattices"
] |
proofwiki-22637 | Category CLat is Subcategory of Frm | Let $\mathbf{Frm}$ denote the category of frames.
Let $\mathbf{CLat}$ denote the category of complete lattices.
Then:
:$\mathbf{CLat}$ is a subcategory of $\mathbf{Frm}$ | From Complete Lattice Homomorphism is Frame Homomorphism:
:$\mathbf{CLat}$ consists of a subcollection of objects and subcollection of morphisms of $\mathbf{Frm}$
By definition of category of complete lattices $\mathbf{CLat}$ and category of frames $\mathbf{Frm}$:
:composition of morphisms in both categories is standar... | Let $\mathbf{Frm}$ denote the [[Definition:Category of Frames|category of frames]].
Let $\mathbf{CLat}$ denote the [[Definition:Category of Complete Lattices|category of complete lattices]].
Then:
:$\mathbf{CLat}$ is a [[Definition:Subcategory|subcategory]] of $\mathbf{Frm}$ | From [[Complete Lattice Homomorphism is Frame Homomorphism]]:
:$\mathbf{CLat}$ consists of a [[Definition:Subcollection|subcollection]] of [[Definition:Object (Category Theory)|objects]] and [[Definition:Subcollection|subcollection]] of [[Definition:Morphism (Category Theory)|morphisms]] of $\mathbf{Frm}$
By definiti... | Category CLat is Subcategory of Frm | https://proofwiki.org/wiki/Category_CLat_is_Subcategory_of_Frm | https://proofwiki.org/wiki/Category_CLat_is_Subcategory_of_Frm | [
"Category of Frames (Lattice Theory)",
"Category of Complete Lattices"
] | [
"Definition:Category of Frames",
"Definition:Category of Complete Lattices",
"Definition:Subcategory"
] | [
"Complete Lattice Homomorphism is Frame Homomorphism",
"Definition:Subcollection",
"Definition:Object (Category Theory)",
"Definition:Subcollection",
"Definition:Morphism",
"Definition:Category of Complete Lattices",
"Definition:Category of Frames",
"Definition:Composition of Morphisms",
"Definition... |
proofwiki-22638 | Adjunction Induces Counit of Adjunction | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $\tuple {F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$.
Let $\beta$ be the inverse of the natural isomorphism $\alpha$.
Then:
:there exists a counit of adjunction $\tuple {F, G, \alpha}$
That is, there exists a natural transformation $... | From Characterization of Adjunction Using Left Adjuncts of Morphisms, we have:
:$(1):\quad$ for every $g:C_1 \to C_2 \in \mathbf C$:
::$\qquad\qquad\map {\beta_{\tuple{GC_1, C_2} } } {G g} = g \circ \map {\beta_{\tuple{GC_1, C_1} } } {\operatorname{id}_{GC_1}}$
:$(2):\quad$ for every $f:D \to GC \in \mathbf D$:
::$\qqu... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $\tuple {F, G, \alpha}$ be an [[Definition:Adjunction|adjunction]] between $\mathbf C$ and $\mathbf D$.
Let $\beta$ be the [[Definition:Inverse Natural Isomorphism between Covariant Functors|inverse]] of the [[Definit... | From [[Characterization of Adjunction Using Left Adjuncts of Morphisms]], we have:
:$(1):\quad$ for every $g:C_1 \to C_2 \in \mathbf C$:
::$\qquad\qquad\map {\beta_{\tuple{GC_1, C_2} } } {G g} = g \circ \map {\beta_{\tuple{GC_1, C_1} } } {\operatorname{id}_{GC_1}}$
:$(2):\quad$ for every $f:D \to GC \in \mathbf D$:
:... | Adjunction Induces Counit of Adjunction | https://proofwiki.org/wiki/Adjunction_Induces_Counit_of_Adjunction | https://proofwiki.org/wiki/Adjunction_Induces_Counit_of_Adjunction | [
"Adjunctions"
] | [
"Definition:Locally Small Category",
"Definition:Adjunction",
"Definition:Natural Isomorphism between Covariant Functors/Inverse",
"Definition:Natural Isomorphism",
"Definition:Counit of Adjunction",
"Definition:Natural Transformation",
"Definition:Identity Functor",
"Definition:Composition of Functor... | [
"Characterization of Adjunction Using Left Adjuncts of Morphisms",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Natural Transformation",
"Definition:Counit of Adjunction",
"Definition:Adjunction"
] |
proofwiki-22639 | Adjunction Induces Unit of Adjunction | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $\tuple {F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$.
Then:
:there exists a unit of adjunction $\tuple {F, G, \alpha}$
That is, there exists a natural transformation $\eta: \operatorname {id}_{\mathbf D} \to GF$ where:
:* for each ob... | From Characterization of Adjunction Using Right Adjuncts of Morphisms:
:$\alpha$ associates with:
::objects $D$ in $\mathbf D$ and $C$ in $\mathbf C$
:a bijection:
::$\alpha_{\tuple{D, C} } : \map {\operatorname{Hom}_{\mathbf C} } {FD, C} \to \map {\operatorname{Hom}_{\mathbf D} } {D, GC}$
:such that:
::$(1)\quad$ for ... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $\tuple {F, G, \alpha}$ be an [[Definition:Adjunction|adjunction]] between $\mathbf C$ and $\mathbf D$.
Then:
:there exists a [[Definition:Unit of Adjunction|unit of adjunction $\tuple {F, G, \alpha}$]]
That is, th... | From [[Characterization of Adjunction Using Right Adjuncts of Morphisms]]:
:$\alpha$ associates with:
::[[Definition:Object (Category Theory)|objects]] $D$ in $\mathbf D$ and $C$ in $\mathbf C$
:a [[Definition:Bijection|bijection]]:
::$\alpha_{\tuple{D, C} } : \map {\operatorname{Hom}_{\mathbf C} } {FD, C} \to \map {\o... | Adjunction Induces Unit of Adjunction | https://proofwiki.org/wiki/Adjunction_Induces_Unit_of_Adjunction | https://proofwiki.org/wiki/Adjunction_Induces_Unit_of_Adjunction | [
"Adjunctions"
] | [
"Definition:Locally Small Category",
"Definition:Adjunction",
"Definition:Unit of Adjunction",
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Definition:Identity Functor",
"Definition:Composition of Functors"
] | [
"Characterization of Adjunction Using Right Adjuncts of Morphisms",
"Definition:Object (Category Theory)",
"Definition:Bijection",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Natural Transformation",
"Definition:Unit of Adjunction",
"Definition:Adjunction"
] |
proofwiki-22640 | Approximation of Lower Darboux Integral by Continuous Function | Let $f : \closedint a b \to \R$ be a bounded real function.
Then, for each $\epsilon > 0$, there is a continuous real function $g : \closedint a b \to \R$ such that:
:$\paren 1 \quad$ For all $x \in \closedint a b$, $\map g x \le \map f x$.
:$\paren 2 \quad$ The infimums of $f$ and $g$ on $\closedint a b$ are the same.... | By definition of lower Darboux integral, there is a subdivision $P$ of $\closedint a b$ such that:
:$\ds \underline {\int_a^b} \map f x \rd x < \map L P + \frac \epsilon 2$
where $\map L P$ denotes the lower Darboux sum of $f$ with respect to $P$.
Write $P = \set {x_0, x_1, \dotsc, x_n}$.
For each $\nu \in \set {1, 2, ... | Let $f : \closedint a b \to \R$ be a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Real Function|real function]].
Then, for each $\epsilon > 0$, there is a [[Definition:Continuous Real Function|continuous real function]] $g : \closedint a b \to \R$ such that:
:$\paren 1 \quad$ For all $x \in \closed... | By definition of [[Definition:Lower Darboux Integral|lower Darboux integral]], there is a [[Definition:Subdivision of Interval|subdivision]] $P$ of $\closedint a b$ such that:
:$\ds \underline {\int_a^b} \map f x \rd x < \map L P + \frac \epsilon 2$
where $\map L P$ denotes the [[Definition:Lower Darboux Sum|lower Darb... | Approximation of Lower Darboux Integral by Continuous Function | https://proofwiki.org/wiki/Approximation_of_Lower_Darboux_Integral_by_Continuous_Function | https://proofwiki.org/wiki/Approximation_of_Lower_Darboux_Integral_by_Continuous_Function | [
"Lower Darboux Integral",
"Lower Darboux Integral"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Definition:Infimum of Mapping/Real-Valued Function",
"Definition:Lower Darboux Integral",
"Definition:Definite Integral/Darboux"
] | [
"Definition:Lower Darboux Integral",
"Definition:Subdivision of Interval",
"Definition:Lower Darboux Sum",
"Definition:Infimum of Mapping/Real-Valued Function",
"Definition:Continuous Real Function",
"Definition:Real Interval/Closed",
"Definition:Infimum of Mapping/Real-Valued Function",
"Definition:S... |
proofwiki-22641 | Arzelà's Dominated Convergence Theorem | For each $n \in \N$, let $f_n : \closedint a b \to \R$ be a real function.
Suppose that each $f_n$ is Darboux integrable on $\closedint a b$.
Further suppose that there is a constant $M > 0$ such that:
:$\size {\map {f_n} x} \le M$
for all $n \in \N$ and $x \in \closedint a b$.
Let $\sequence {f_n}$ converge pointwise ... | {{ProofWanted}}
Category:Definite Integrals
g0yet0yfgzomimlfg1r82z1xna0aegm | For each $n \in \N$, let $f_n : \closedint a b \to \R$ be a [[Definition:Real Function|real function]].
Suppose that each $f_n$ is [[Definition:Darboux Integrable Function|Darboux integrable]] on $\closedint a b$.
Further suppose that there is a constant $M > 0$ such that:
:$\size {\map {f_n} x} \le M$
for all $n \in... | {{ProofWanted}}
[[Category:Definite Integrals]]
g0yet0yfgzomimlfg1r82z1xna0aegm | Arzelà's Dominated Convergence Theorem | https://proofwiki.org/wiki/Arzelà's_Dominated_Convergence_Theorem | https://proofwiki.org/wiki/Arzelà's_Dominated_Convergence_Theorem | [
"Definite Integrals"
] | [
"Definition:Real Function",
"Definition:Darboux Integrable Function",
"Definition:Pointwise Convergence",
"Definition:Real Function",
"Definition:Darboux Integrable Function"
] | [
"Category:Definite Integrals"
] |
proofwiki-22642 | Complex Conjugation is Real Vector Space Automorphism | The operation of complex conjugation:
:$\forall z \in \C: z \mapsto \overline z$
is a vector space automorphism over the real vector space $\R^n$. | {{ProofWanted|first define exactly what the above means -- it gets a one-liner in the given source}} | The operation of [[Definition:Complex Conjugate|complex conjugation]]:
:$\forall z \in \C: z \mapsto \overline z$
is a [[Definition:Vector Space Automorphism|vector space automorphism]] over the [[Definition:Real Vector Space|real vector space]] $\R^n$. | {{ProofWanted|first define exactly what the above means -- it gets a one-liner in the given source}} | Complex Conjugation is Real Vector Space Automorphism | https://proofwiki.org/wiki/Complex_Conjugation_is_Real_Vector_Space_Automorphism | https://proofwiki.org/wiki/Complex_Conjugation_is_Real_Vector_Space_Automorphism | [
"Complex Conjugates",
"Vector Space Automorphisms",
"Real Vector Spaces"
] | [
"Definition:Complex Conjugate",
"Definition:Vector Space Automorphism",
"Definition:Real Vector Space"
] | [] |
proofwiki-22643 | Complex Conjugation is not Complex Vector Space Automorphism | The operation of complex conjugation:
:$\forall z \in \C: z \mapsto \overline z$
is ''not'' a vector space automorphism over the complex vector space $\C^n$. | {{ProofWanted|first define exactly what the above means -- it gets a one-liner in the given source}} | The operation of [[Definition:Complex Conjugate|complex conjugation]]:
:$\forall z \in \C: z \mapsto \overline z$
is ''not'' a [[Definition:Vector Space Automorphism|vector space automorphism]] over the [[Definition:Complex Vector Space|complex vector space]] $\C^n$. | {{ProofWanted|first define exactly what the above means -- it gets a one-liner in the given source}} | Complex Conjugation is not Complex Vector Space Automorphism | https://proofwiki.org/wiki/Complex_Conjugation_is_not_Complex_Vector_Space_Automorphism | https://proofwiki.org/wiki/Complex_Conjugation_is_not_Complex_Vector_Space_Automorphism | [
"Complex Conjugates",
"Vector Space Automorphisms",
"Complex Vector Spaces"
] | [
"Definition:Complex Conjugate",
"Definition:Vector Space Automorphism",
"Definition:Complex Vector Space"
] | [] |
proofwiki-22644 | Empty Topological Space is T2 | Let $T = \struct {\O, \set \O}$ be the empty topological space.
Then $T$ is a $T_2$ (Hausdorff) space. | {{Recall|T2 Space|$T_2$ (Hausdorff) space}}
{{:Definition:T2 Space/Definition 1}}
This is vacuously true for the empty set.
{{qed}}
Category:Empty Topological Space
Category:Examples of Hausdorff Spaces
6jtisofefvxrvtvg7o7n5hczeatzjsx | Let $T = \struct {\O, \set \O}$ be the [[Definition:Empty Topological Space|empty topological space]].
Then $T$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | {{Recall|T2 Space|$T_2$ (Hausdorff) space}}
{{:Definition:T2 Space/Definition 1}}
This is [[Definition:Vacuous Truth|vacuously true]] for the [[Definition:Empty Set|empty set]].
{{qed}}
[[Category:Empty Topological Space]]
[[Category:Examples of Hausdorff Spaces]]
6jtisofefvxrvtvg7o7n5hczeatzjsx | Empty Topological Space is T2 | https://proofwiki.org/wiki/Empty_Topological_Space_is_T2 | https://proofwiki.org/wiki/Empty_Topological_Space_is_T2 | [
"Empty Topological Space",
"Examples of Hausdorff Spaces"
] | [
"Definition:Empty Topological Space",
"Definition:T2 Space"
] | [
"Definition:Vacuous Truth",
"Definition:Empty Set",
"Category:Empty Topological Space",
"Category:Examples of Hausdorff Spaces"
] |
proofwiki-22645 | Empty Topological Space is Second Countable | Let $T = \struct {\O, \set \O}$ be the empty topological space.
Then $T$ is second-countable. | Follows from Finite Space is Second-Countable.
{{qed}}
Category:Empty Topological Space
9af0xahl6fvkn2jao68vfl5eh1u9uh9 | Let $T = \struct {\O, \set \O}$ be the [[Definition:Empty Topological Space|empty topological space]].
Then $T$ is [[Definition:Second-Countable Space|second-countable]]. | Follows from [[Finite Space is Second-Countable]].
{{qed}}
[[Category:Empty Topological Space]]
9af0xahl6fvkn2jao68vfl5eh1u9uh9 | Empty Topological Space is Second Countable | https://proofwiki.org/wiki/Empty_Topological_Space_is_Second_Countable | https://proofwiki.org/wiki/Empty_Topological_Space_is_Second_Countable | [
"Empty Topological Space",
"Empty Topological Space"
] | [
"Definition:Empty Topological Space",
"Definition:Second-Countable Space"
] | [
"Finite Space is Second-Countable",
"Category:Empty Topological Space"
] |
proofwiki-22646 | Empty Topological Space is Locally Euclidean Space of any Dimension | Let $T = \struct {\O, \set \O}$ be the empty topological space.
Then $T$ is locally euclidean space for every dimension. | {{Recall|Locally Euclidean Space}}
{{:Definition:Locally Euclidean Space}}
This is vacuously true for the empty set for every $d \in \N$.
{{qed}}
Category:Empty Topological Space
Category:Locally Euclidean Spaces
h2trl85kjmoc7n9ydfcah3mi5u7xc61 | Let $T = \struct {\O, \set \O}$ be the [[Definition:Empty Topological Space|empty topological space]].
Then $T$ is [[Definition:Locally Euclidean Space|locally euclidean space]] for every [[Definition:Dimension of Locally Euclidean Space|dimension]]. | {{Recall|Locally Euclidean Space}}
{{:Definition:Locally Euclidean Space}}
This is [[Definition:Vacuous Truth|vacuously true]] for the [[Definition:Empty Set|empty set]] for every $d \in \N$.
{{qed}}
[[Category:Empty Topological Space]]
[[Category:Locally Euclidean Spaces]]
h2trl85kjmoc7n9ydfcah3mi5u7xc61 | Empty Topological Space is Locally Euclidean Space of any Dimension | https://proofwiki.org/wiki/Empty_Topological_Space_is_Locally_Euclidean_Space_of_any_Dimension | https://proofwiki.org/wiki/Empty_Topological_Space_is_Locally_Euclidean_Space_of_any_Dimension | [
"Empty Topological Space",
"Locally Euclidean Spaces"
] | [
"Definition:Empty Topological Space",
"Definition:Locally Euclidean Space",
"Definition:Dimension (Topology)/Locally Euclidean Space"
] | [
"Definition:Vacuous Truth",
"Definition:Empty Set",
"Category:Empty Topological Space",
"Category:Locally Euclidean Spaces"
] |
proofwiki-22647 | Baire Category Theorem/Hausdorff Space | Let $T = \struct {S, \tau}$ be a Hausdorff space.
Let $T$ be locally compact.
Then $T = \struct {S, \tau}$ is also a Baire space. | {{ProofWanted}}
{{Namedfor|René-Louis Baire|cat = Baire}} | Let $T = \struct {S, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $T$ be [[Definition:Locally Compact Space|locally compact]].
Then $T = \struct {S, \tau}$ is also a [[Definition:Baire Space (Topology)|Baire space]]. | {{ProofWanted}}
{{Namedfor|René-Louis Baire|cat = Baire}} | Baire Category Theorem/Hausdorff Space | https://proofwiki.org/wiki/Baire_Category_Theorem/Hausdorff_Space | https://proofwiki.org/wiki/Baire_Category_Theorem/Hausdorff_Space | [
"Baire Category Theorem",
"Hausdorff Spaces",
"Locally Compact Spaces",
"Baire Spaces"
] | [
"Definition:T2 Space",
"Definition:Locally Compact Space",
"Definition:Baire Space (Topology)"
] | [] |
proofwiki-22648 | Euclidean Space is Second-Countable | Let $\R^n$ be an $n$-dimensional real vector space.
Let $\struct {\R^n, \tau_d}$ $n$-dimensional real Euclidean space with the usual topology.
Then, $\struct {\R^n, \tau_d}$ is a second-countable space. | Let $\struct {\R, \tau}$ be the real line with usual topology.
Let $T_n = \struct {\R^n, \tau_n}$ be the topological space such that $\tau_n$ is the product topology on $\R^n$.
From Real Number Line is Second-Countable, we have that $\struct {\R, \tau}$ is second-countable.
From Countable Product of Second-Countable Sp... | Let $\R^n$ be an [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Real Vector Space|real vector space]].
Let $\struct {\R^n, \tau_d}$ [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Real Euclidean n-Space|real Euclidean space with the usual topology]].
Then, $\struct {\R^n,... | Let $\struct {\R, \tau}$ be the [[Definition:Real Number Line with Euclidean Topology|real line with usual topology]].
Let $T_n = \struct {\R^n, \tau_n}$ be the [[Definition:Topological Space|topological space]] such that $\tau_n$ is the [[Definition:Product Topology|product topology]] on $\R^n$.
From [[Real Number ... | Euclidean Space is Second-Countable | https://proofwiki.org/wiki/Euclidean_Space_is_Second-Countable | https://proofwiki.org/wiki/Euclidean_Space_is_Second-Countable | [
"Real Euclidean Spaces",
"Examples of Second-Countable Spaces"
] | [
"Definition:Dimension of Vector Space",
"Definition:Real Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Euclidean Space/Euclidean Topology/Real",
"Definition:Second-Countable Space"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Topological Space",
"Definition:Product Topology",
"Real Number Line is Second-Countable",
"Definition:Second-Countable Space",
"Countable Product of Second-Countable Spaces is Second-Countable",
"Definition:Second-Countable S... |
proofwiki-22649 | Empty Topological Space is n-Manifold | Let $T = \struct {\O, \set \O}$ be the empty topological space.
Then $T$ is a topological $n$-manifold for every $n \in \N$. | {{Recall|Topological Manifold|topological $d$-manifold}}
{{:Definition:Topological Manifold}}
From Empty Topological Space is $T_2$, $T$ is a $T_2$ space.
From Empty Topological Space is Second Countable, $T$ is second-countable.
From Empty Topological Space is Locally Euclidean Space of any Dimension, $T$ is a locally... | Let $T = \struct {\O, \set \O}$ be the [[Definition:Empty Topological Space|empty topological space]].
Then $T$ is a [[Definition:Topological Manifold|topological $n$-manifold]] for every $n \in \N$. | {{Recall|Topological Manifold|topological $d$-manifold}}
{{:Definition:Topological Manifold}}
From [[Empty Topological Space is T2|Empty Topological Space is $T_2$]], $T$ is a [[Definition:T2 Space|$T_2$ space]].
From [[Empty Topological Space is Second Countable]], $T$ is [[Definition:Second-Countable Space|second-... | Empty Topological Space is n-Manifold | https://proofwiki.org/wiki/Empty_Topological_Space_is_n-Manifold | https://proofwiki.org/wiki/Empty_Topological_Space_is_n-Manifold | [
"Empty Topological Space",
"Examples of Topological Manifolds"
] | [
"Definition:Empty Topological Space",
"Definition:Topological Manifold"
] | [
"Empty Topological Space is T2",
"Definition:T2 Space",
"Empty Topological Space is Second Countable",
"Definition:Second-Countable Space",
"Empty Topological Space is Locally Euclidean Space of any Dimension",
"Definition:Locally Euclidean Space",
"Definition:Topological Manifold",
"Category:Empty To... |
proofwiki-22650 | Interior of Topological n-Manifold with Boundary is Topological n-Manifold without Boundary | Let $M$ be a topological $n$-manifold with boundary.
The '''interior''' of $M$, denoted $\Int M$, is a topological $n$-manifold (without boundary). | {{Recall|Locally Euclidean Space|locally Euclidean space}}
{{:Definition:Locally Euclidean Space}}
{{Recall|Topological Manifold|manifold (without boundary)}}
{{:Definition:Topological Manifold}}
{{Recall|Topological Manifold with Boundary|manifold with boundary and its interior}}
{{:Definition:Topological Manifold wit... | Let $M$ be a [[Definition:Topological Manifold with Boundary|topological $n$-manifold with boundary]].
The '''[[Definition:Interior of Topological Manifold with Boundary|interior]]''' of $M$, denoted $\Int M$, is a [[Definition:Topological Manifold|topological $n$-manifold (without boundary)]]. | {{Recall|Locally Euclidean Space|locally Euclidean space}}
{{:Definition:Locally Euclidean Space}}
{{Recall|Topological Manifold|manifold (without boundary)}}
{{:Definition:Topological Manifold}}
{{Recall|Topological Manifold with Boundary|manifold with boundary and its interior}}
{{:Definition:Topological Manifold w... | Interior of Topological n-Manifold with Boundary is Topological n-Manifold without Boundary | https://proofwiki.org/wiki/Interior_of_Topological_n-Manifold_with_Boundary_is_Topological_n-Manifold_without_Boundary | https://proofwiki.org/wiki/Interior_of_Topological_n-Manifold_with_Boundary_is_Topological_n-Manifold_without_Boundary | [
"Definitions/Topological Manifolds with Boundary",
"Definitions/Topological Manifolds"
] | [
"Definition:Topological Manifold with Boundary",
"Definition:Topological Manifold with Boundary/Interior",
"Definition:Topological Manifold"
] | [
"Definition:Second-Countable Space",
"Definition:T2 Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Open Set/Topology",
"Definition:Euclidean Space",
"Definition:Topological Manifold"
] |
proofwiki-22651 | Barycentric Coordinates of Centroid of Triangle | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be the position vectors of the $3$ vertices of a triangle $T$ in the plane.
Let $p$ be the centroid of $T$.
Let $\alpha$, $\beta$ and $\gamma$ be the barycentric coordinates of $p$ {{WRT}} $T$
Then:
:$\alpha = \beta = \gamma = \dfrac 1 3$ | thumbalt=A triangle with vertices given by position vectors a, b, and c. The midpoint of bc is marked as q. On the median line aq, the median of the triangle is marked at point p.A [[Definition:Triangletriangle with vertices given by position vectors $\mathbf a$, $\mathbf b$, and $\mathbf c$. The midpoint of $\overline... | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be the [[Definition:Position Vector|position vectors]] of the $3$ [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Triangle (Geometry)|triangle]] $T$ in [[Definition:The Plane|the plane]].
Let $p$ be the [[Definition:Centroid of Triangle|centroid]] of $T$.
Let $... | [[File:TriangleMedianPoint.png|thumb|alt=A triangle with vertices given by position vectors a, b, and c. The midpoint of bc is marked as q. On the median line aq, the median of the triangle is marked at point p.|A [[Definition:Triangle|triangle]] with vertices given by [[Definition:Position Vector|position vectors]] $\... | Barycentric Coordinates of Centroid of Triangle | https://proofwiki.org/wiki/Barycentric_Coordinates_of_Centroid_of_Triangle | https://proofwiki.org/wiki/Barycentric_Coordinates_of_Centroid_of_Triangle | [
"Barycentric Coordinates"
] | [
"Definition:Position Vector",
"Definition:Polygon/Vertex",
"Definition:Triangle (Geometry)",
"Definition:Plane Surface/The Plane",
"Definition:Centroid/Triangle",
"Definition:Barycentric Coordinates"
] | [
"File:TriangleMedianPoint.png",
"Definition:Triangle",
"Definition:Position Vector",
"Definition:Midpoint",
"Definition:Median of Triangle",
"Definition:Median of Triangle",
"Definition:Position Vector",
"Vector Equation of Straight Line in Space/Formulation 2",
"Definition:Position Vector",
"Defi... |
proofwiki-22652 | Ideal and Zero Locus are Order Reversing | Let $k$ be a field.
Let $n \ge 0$ be a natural number.
Let $P := k \sqbrk {x_1, \ldots, x_n}$ be the polynomial ring in $n$ variables over $k$.
Then, the associate ideal and zero-locus are inclusion reversing.
That is:
:$(1): \quad$ if:
::::$X \subseteq Y \subseteq k^n$
:::then:
::::$\map I Y \subseteq \map I X$
:$(2):... | ;<nowiki>$(1):$</nowiki> zero-locus is inclusion reversing
By definition, the '''zero locus''' of $I$ is the set:
:$\map V I = \set {x \in k^n : \forall f \in I: \map f x = 0}$
Let $X \subseteq Y \subseteq k^n$ be nested sets.
Let $p \in k \sqbrk {x_1, \ldots, x_n}$ be a polynomial in $k$ such that $p$ vanishes on $Y$... | Let $k$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $n \ge 0$ be a [[Definition:Natural Number|natural number]].
Let $P := k \sqbrk {x_1, \ldots, x_n}$ be the [[Definition:Polynomial Ring in Multiple Variables|polynomial ring]] in $n$ variables over $k$.
Then, the [[Definition:Associated Ideal of Subset ... | ;<nowiki>$(1):$</nowiki> [[Definition:Zero Locus of Set of Polynomials|zero-locus]] is [[Definition:Inclusion-Reversing Mapping|inclusion reversing]]
By definition, the '''[[Definition:Zero Locus of Set of Polynomials|zero locus]]''' of $I$ is the [[Definition:Set|set]]:
:$\map V I = \set {x \in k^n : \forall f \in I... | Ideal and Zero Locus are Order Reversing | https://proofwiki.org/wiki/Ideal_and_Zero_Locus_are_Order_Reversing | https://proofwiki.org/wiki/Ideal_and_Zero_Locus_are_Order_Reversing | [
"Algebraic Geometry"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Natural Numbers",
"Definition:Polynomial Ring",
"Definition:Vanishing Ideal of Subset of Affine Space",
"Definition:Zero Locus of Set of Polynomials",
"Definition:Inclusion-Reversing Mapping",
"Definition:Ideal of Ring"
] | [
"Definition:Zero Locus of Set of Polynomials",
"Definition:Inclusion-Reversing Mapping",
"Definition:Zero Locus of Set of Polynomials",
"Definition:Set",
"Definition:Nested Sets",
"Definition:Polynomial",
"Definition:Vanish",
"Definition:Vanish",
"Definition:Almost Everywhere",
"Definition:Vanishi... |
proofwiki-22653 | Bellman's Principle of Optimality | Let $P$ be an optimal path in an exercise in dynamic programming.
Then any part of $P$ is also an optimal path. | {{ProofWanted}}
{{Namedfor|Richard Ernest Bellman|cat = Bellman}} | Let $P$ be an [[Definition:Optimal Path|optimal path]] in an exercise in [[Definition:Dynamic Programming|dynamic programming]].
Then any part of $P$ is also an [[Definition:Optimal Path|optimal path]]. | {{ProofWanted}}
{{Namedfor|Richard Ernest Bellman|cat = Bellman}} | Bellman's Principle of Optimality | https://proofwiki.org/wiki/Bellman's_Principle_of_Optimality | https://proofwiki.org/wiki/Bellman's_Principle_of_Optimality | [
"Dynamic Programming"
] | [
"Definition:Optimal Path",
"Definition:Dynamic Programming",
"Definition:Optimal Path"
] | [] |
proofwiki-22654 | Benford's Law | '''Benford's law''' is an empirical observation concerning the leading digits in a given number base (usually $10$).
In a data set containing data spanning several orders of magnitude, the leading digits are more likely to be small. | {{ProofWanted}}
{{Namedfor|Frank Albert Benford|cat = Benford}} | '''[[Benford's Law|Benford's law]]''' is an empirical observation concerning the leading [[Definition:Digit|digits]] in a [[Definition:Given|given]] [[Definition:Number Base|number base]] (usually $10$).
In a [[Definition:Data Set|data set]] containing data spanning several [[Definition:Order of Magnitude|orders of ma... | {{ProofWanted}}
{{Namedfor|Frank Albert Benford|cat = Benford}} | Benford's Law | https://proofwiki.org/wiki/Benford's_Law | https://proofwiki.org/wiki/Benford's_Law | [
"Statistics"
] | [
"Benford's Law",
"Definition:Digit",
"Definition:Given",
"Definition:Number Base",
"Definition:Data Set",
"Definition:Order of Magnitude",
"Definition:Digit"
] | [] |
proofwiki-22655 | Direct Image Mapping of Identity is Identity | Let $S$ be a set.
Let $I_S : S \to S$ be the identity mapping on $S$.
Let $\paren{I_S}^\to : \powerset S \to \powerset S$ be the direct image mapping of $I_S$.
Then:
:$\paren{I_S}^\to = I_{\powerset S}$
where $I_{\powerset S} :\powerset S \to \powerset S$ denotes the identity mapping on $\powerset S$. | Let $A \in \powerset S$.
We have:
{{begin-eqn}}
{{eqn | q = \forall A \in \powerset S
| l = \map {\paren{I_S}^\to} A
| r = \set{x \in S : \exists y \in A : \map {I_S} y = x}
| c = {{Defof|Direct Image Mapping}}
}}
{{eqn | ll = \leadsto
| q = \forall A \in \powerset S
| l = A
| o = \s... | Let $S$ be a [[Definition:Set|set]].
Let $I_S : S \to S$ be the [[Definition:Identity Mapping|identity mapping]] on $S$.
Let $\paren{I_S}^\to : \powerset S \to \powerset S$ be the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $I_S$.
Then:
:$\paren{I_S}^\to = I_{\powerset S}$
where $I_{\powe... | Let $A \in \powerset S$.
We have:
{{begin-eqn}}
{{eqn | q = \forall A \in \powerset S
| l = \map {\paren{I_S}^\to} A
| r = \set{x \in S : \exists y \in A : \map {I_S} y = x}
| c = {{Defof|Direct Image Mapping}}
}}
{{eqn | ll = \leadsto
| q = \forall A \in \powerset S
| l = A
| o = ... | Direct Image Mapping of Identity is Identity | https://proofwiki.org/wiki/Direct_Image_Mapping_of_Identity_is_Identity | https://proofwiki.org/wiki/Direct_Image_Mapping_of_Identity_is_Identity | [
"Direct Image Mappings",
"Inverse Mappings"
] | [
"Definition:Set",
"Definition:Identity Mapping",
"Definition:Direct Image Mapping/Mapping",
"Definition:Identity Mapping"
] | [
"Relative Complement inverts Subsets",
"Equality of Mappings",
"Category:Direct Image Mappings",
"Category:Inverse Mappings"
] |
proofwiki-22656 | Inverse of Direct Image Mapping is Direct Image Mapping of Inverse | Let $S$ and $T$ be sets.
Let $f :S \to T$ be a bijection.
Let $f^\to$ be the direct image mapping of $f$.
Then:
:$\paren {f^\to}^{-1} = \paren {f^{-1} }^{\to}$
where $f^{-1}$ and $\paren{f^\to}^{-1}$ denote the inverses of $f$ and $f^\to$ respectively | From Mapping is Bijection iff Direct Image Mapping is Bijection:
:$f^\to$ is a bijection
From Bijection has Left and Right Inverse:
:the inverses $f^{-1}$ and $\paren{f^\to}^{-1}$ are well-defined
We have:
{{begin-eqn}}
{{eqn | l = f^\to \circ \paren{f^{-1} }^\to
| r = \paren{f \circ f^{-1} }^\to
| c = Comp... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f :S \to T$ be a [[Definition:Bijection|bijection]].
Let $f^\to$ be the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $f$.
Then:
:$\paren {f^\to}^{-1} = \paren {f^{-1} }^{\to}$
where $f^{-1}$ and $\paren{f^\to}^{-1}$ denote the [[Definition:I... | From [[Mapping is Bijection iff Direct Image Mapping is Bijection]]:
:$f^\to$ is a [[Definition:Bijection|bijection]]
From [[Bijection has Left and Right Inverse]]:
:the [[Definition:Inverse Mapping|inverses]] $f^{-1}$ and $\paren{f^\to}^{-1}$ are [[Definition:Well-Defined|well-defined]]
We have:
{{begin-eqn}}
{{eqn... | Inverse of Direct Image Mapping is Direct Image Mapping of Inverse | https://proofwiki.org/wiki/Inverse_of_Direct_Image_Mapping_is_Direct_Image_Mapping_of_Inverse | https://proofwiki.org/wiki/Inverse_of_Direct_Image_Mapping_is_Direct_Image_Mapping_of_Inverse | [
"Direct Image Mappings",
"Identity Mappings"
] | [
"Definition:Set",
"Definition:Bijection",
"Definition:Direct Image Mapping/Mapping",
"Definition:Inverse of Mapping"
] | [
"Mapping is Bijection iff Direct Image Mapping is Bijection",
"Definition:Bijection",
"Bijection has Left and Right Inverse",
"Definition:Inverse Mapping",
"Definition:Well-Defined",
"Composition of Direct Image Mappings of Mappings",
"Direct Image Mapping of Identity is Identity",
"Definition:Inverse... |
proofwiki-22657 | Identity Mapping is Frame Isomorphism | Let $L = \struct{S, \preceq}$ be a frame.
Let $\operatorname{id}_S$ denote the identity mapping on $S$.
Then:
:$\operatorname{id}_S$ is a frame isomorphism of $L$ to $L$ | From Identity Mapping is Bijection and Identity Mapping is Frame Homomorphism:
:$\operatorname{id}_S$ is a bijective frame homomorphism
Hence $\operatorname{id}_S$ is a frame isomorphism by definition.
{{qed}}
Category:Frame Isomorphisms
h9ytnyjd9b6yltuwrq43smfjxj4lwlx | Let $L = \struct{S, \preceq}$ be a [[Definition:Frame (Lattice Theory)|frame]].
Let $\operatorname{id}_S$ denote the [[Definition:Identity Mapping|identity mapping]] on $S$.
Then:
:$\operatorname{id}_S$ is a [[Definition:Frame Isomorphism|frame isomorphism]] of $L$ to $L$ | From [[Identity Mapping is Bijection]] and [[Identity Mapping is Frame Homomorphism]]:
:$\operatorname{id}_S$ is a [[Definition:Bijection|bijective]] [[Definition:Frame Homomorphism|frame homomorphism]]
Hence $\operatorname{id}_S$ is a [[Definition:Frame Isomorphism|frame isomorphism]] by definition.
{{qed}}
[[Categ... | Identity Mapping is Frame Isomorphism | https://proofwiki.org/wiki/Identity_Mapping_is_Frame_Isomorphism | https://proofwiki.org/wiki/Identity_Mapping_is_Frame_Isomorphism | [
"Frame Isomorphisms"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Identity Mapping",
"Definition:Frame Isomorphism"
] | [
"Identity Mapping is Bijection",
"Identity Mapping is Frame Homomorphism",
"Definition:Bijection",
"Definition:Frame Homomorphism",
"Definition:Frame Isomorphism",
"Category:Frame Isomorphisms"
] |
proofwiki-22658 | Convergent Sequence with Finite Number of Terms Inserted is Convergent | Let $\struct {X, d}$ be a metric space.
Let $\sequence {x_k}$ be a sequence in $X$.
Let $\sequence {x_k}$ be convergent.
Let a finite number of terms be inserted into $\sequence {x_k}$.
Then the resulting sequence is convergent. | Let $I$ be the index of the last inserted term.
Let $N$ be the length of the sequence of terms to be inserted.
By Convergent Sequence with Finite Number of Terms Deleted is Convergent, we may delete the first $I$ terms from the original sequence $\sequence {x_k}$.
Let $\sequence {f_k}$ be the sequence of those $I$ term... | Let $\struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {x_k}$ be a [[Definition:Sequence|sequence in $X$]].
Let $\sequence {x_k}$ be [[Definition:Convergent Sequence (Metric Space)|convergent]].
Let a [[Definition:Finite Set|finite]] number of [[Definition:Term of Sequence|terms]] be ins... | Let $I$ be the [[Definition:Index of Term of Sequence|index]] of the last inserted [[Definition:Term of Sequence|term]].
Let $N$ be the [[Definition:Length of Sequence|length]] of the [[Definition:Sequence|sequence]] of [[Definition:Term of Sequence|terms]] to be inserted.
By [[Convergent Sequence with Finite Number ... | Convergent Sequence with Finite Number of Terms Inserted is Convergent | https://proofwiki.org/wiki/Convergent_Sequence_with_Finite_Number_of_Terms_Inserted_is_Convergent | https://proofwiki.org/wiki/Convergent_Sequence_with_Finite_Number_of_Terms_Inserted_is_Convergent | [] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Finite Set",
"Definition:Term of Sequence",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space"
] | [
"Definition:Term of Sequence/Index",
"Definition:Term of Sequence",
"Definition:Length of Sequence",
"Definition:Sequence",
"Definition:Term of Sequence",
"Convergent Sequence with Finite Number of Terms Deleted is Convergent",
"Definition:Term of Sequence",
"Definition:Sequence",
"Definition:Sequen... |
proofwiki-22659 | Outer Jordan Content of Separated Union | Let $M \subseteq \R^n$ be a bounded subset of Euclidean $n$-space.
Fix $k \in \set {1, 2, \dots, n}$ and $\gamma \in \R$.
For $\vec x \in \R^n$, let $\vec x = \tuple {x_1, x_2, \dots, x_n}$
Let:
:$A = \set {\vec x \in M : x_k < \gamma}$
:$B = \set {\vec x \in M : x_k > \gamma}$
Then:
:$\map {m^*} M = \map {m^*} A + \ma... | Let $C = \set {\vec x \in M : x_k = \gamma}$.
By Trichotomy Law for Real Numbers:
:$M = A \cup B \cup C$
and so by Outer Jordan Content is Subadditive:
:$\map {m^*} M \le \map {m^*} A + \map {m^*} B + \map {m^*} C$
But by Outer Jordan Content of Degenerate Set:
:$\map {m^*} C = 0$
and so:
:$\map {m^*} M \le \map {m^*} ... | Let $M \subseteq \R^n$ be a [[Definition:Bounded Euclidean Space|bounded]] [[Definition:Subset|subset]] of [[Definition:Real Euclidean Space|Euclidean $n$-space]].
Fix $k \in \set {1, 2, \dots, n}$ and $\gamma \in \R$.
For $\vec x \in \R^n$, let $\vec x = \tuple {x_1, x_2, \dots, x_n}$
Let:
:$A = \set {\vec x \in M... | Let $C = \set {\vec x \in M : x_k = \gamma}$.
By [[Trichotomy Law for Real Numbers]]:
:$M = A \cup B \cup C$
and so by [[Outer Jordan Content is Subadditive]]:
:$\map {m^*} M \le \map {m^*} A + \map {m^*} B + \map {m^*} C$
But by [[Outer Jordan Content of Degenerate Set]]:
:$\map {m^*} C = 0$
and so:
:$\map {m^*} M \... | Outer Jordan Content of Separated Union | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Separated_Union | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Separated_Union | [
"Outer Jordan Content"
] | [
"Definition:Bounded Metric Space/Euclidean",
"Definition:Subset",
"Definition:Euclidean Space/Real",
"Definition:Outer Jordan Content"
] | [
"Trichotomy Law for Real Numbers",
"Outer Jordan Content is Subadditive",
"Outer Jordan Content of Degenerate Set",
"Characterizing Property of Infimum of Subset of Real Numbers",
"Definition:Cover of Set/Finite",
"Definition:Closed Rectangle",
"Trichotomy Law for Real Numbers",
"Definition:Set Partit... |
proofwiki-22660 | Transfer Operator on Lipshitz Functions on Shift Space is Linear Bounded Operator | Let $\struct {X ^+, \sigma}$ be a one-sided shift of finite type.
Let $F_\theta^+$ be the space of Lipschitz functions with the Lipschitz norm $\norm \cdot_\theta$.
Let $f \in F_\theta^+$.
Let $\LL_f : F_\theta^+ \to F_\theta^+$ be the transfer operator
Then $\LL_f$ is a bounded linear operator. | Let $\mathbf A$ be the $k\times k$ logical matrix of $X ^+$.
Let $n \ge 1$.
Let $x, y \in X^+$ be such that:
:$\forall i \in \hointr 0 n : x_i = y_i$
Then:
{{begin-eqn}}
{{eqn | l = \cmod {\map {\LL_f g} x - \map {\LL_f g} y}
| r = \cmod {\sum_{\map {\mathbf A} {i, x_0} = 1} e^{\map f {ix} } \map g {ix} - \sum_{\... | Let $\struct {X ^+, \sigma}$ be a [[Definition:One-Sided Shift of Finite Type|one-sided shift of finite type]].
Let $F_\theta^+$ be the [[Definition:Space of Lipschitz Functions/One-Sided Shift of Finite Type|space of Lipschitz functions]] with the [[Definition:Lipschitz Norm|Lipschitz norm]] $\norm \cdot_\theta$.
Le... | Let $\mathbf A$ be the $k\times k$ [[Definition:Logical Matrix|logical]] [[Definition:Matrix|matrix]] of $X ^+$.
Let $n \ge 1$.
Let $x, y \in X^+$ be such that:
:$\forall i \in \hointr 0 n : x_i = y_i$
Then:
{{begin-eqn}}
{{eqn | l = \cmod {\map {\LL_f g} x - \map {\LL_f g} y}
| r = \cmod {\sum_{\map {\mathbf ... | Transfer Operator on Lipshitz Functions on Shift Space is Linear Bounded Operator | https://proofwiki.org/wiki/Transfer_Operator_on_Lipshitz_Functions_on_Shift_Space_is_Linear_Bounded_Operator | https://proofwiki.org/wiki/Transfer_Operator_on_Lipshitz_Functions_on_Shift_Space_is_Linear_Bounded_Operator | [
"Ergodic Theory",
"Functional Analysis"
] | [
"Definition:One-Sided Shift of Finite Type",
"Definition:Space of Lipschitz Functions/One-Sided Shift of Finite Type",
"Definition:Lipschitz Norm",
"Definition:Ruelle-Perron-Frobenius Operator/One-Sided Shift Space of Finite Type",
"Definition:Bounded Linear Operator/Normed Vector Space"
] | [
"Definition:Logical Matrix",
"Definition:Matrix",
"Mean Value Theorem",
"Transfer Operator with respect to One-Sided Shift Space of Finite Type is Linear Bounded Operator",
"Definition:Norm/Bounded Linear Transformation",
"Category:Ergodic Theory",
"Category:Functional Analysis"
] |
proofwiki-22661 | Borel-Cantelli Lemma in Probability | Let $\sequence {E_n}$ be an arbitrary countable sequence of events.
Let the sum of the probabilities of $\sequence {E_n}$ be finite.
Then the probability that infinitely many of the events occur is zero. | {{ProofWanted}}
{{Namedfor|Émile Borel|name2 = Francesco Paolo Cantelli|cat = Borel, Émile|cat2 = Cantelli}} | Let $\sequence {E_n}$ be an [[Definition:Arbitrary|arbitrary]] [[Definition:Countable Set|countable]] [[Definition:Sequence|sequence]] of [[Definition:Event|events]].
Let the [[Definition:Sum|sum]] of the [[Definition:Probability|probabilities]] of $\sequence {E_n}$ be [[Definition:Finite|finite]].
Then the [[Defini... | {{ProofWanted}}
{{Namedfor|Émile Borel|name2 = Francesco Paolo Cantelli|cat = Borel, Émile|cat2 = Cantelli}} | Borel-Cantelli Lemma in Probability | https://proofwiki.org/wiki/Borel-Cantelli_Lemma_in_Probability | https://proofwiki.org/wiki/Borel-Cantelli_Lemma_in_Probability | [
"Borel-Cantelli Lemma"
] | [
"Definition:Arbitrary",
"Definition:Countable Set",
"Definition:Sequence",
"Definition:Event",
"Definition:Sum",
"Definition:Probability",
"Definition:Finite",
"Definition:Probability",
"Definition:Infinite Set",
"Definition:Event",
"Definition:Event/Occurrence",
"Definition:Zero (Number)"
] | [] |
proofwiki-22662 | Contravariant Functor Induces Covariant Functor from Dual Category | Let $\mathbf C$ and $\mathbf D$ be metacategories.
Let $F : \mathbf C \to \mathbf D$ be a contravariant functor.
Let $\mathbf C^{\text {op}}$ denote the dual category of $\mathbf C$.
Then $F': \mathbf C^{\text {op}} \to \mathbf D$ defined by:
:* each object $C^{\text{op}}$ of $\mathbf C^{\text{op}}$ is assigned to the ... | We have:
{{begin-eqn}}
{{eqn | l = \map {F'} {g^{\text {op} } \circ f^{\text {op} } }
| r = \map {F'} {\paren{f \circ g }^{\text {op} } }
| c = {{Defof|Dual Category}}
}}
{{eqn | r = \map F {f \circ g}
| c = Definition of $F'$
}}
{{eqn | r = \map F g \circ \map F f
| c = {{Defof|Contravariant F... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Metacategory|metacategories]].
Let $F : \mathbf C \to \mathbf D$ be a [[Definition:Contravariant Functor|contravariant functor]].
Let $\mathbf C^{\text {op}}$ denote the [[Definition:Dual Category|dual category]] of $\mathbf C$.
Then $F': \mathbf C^{\text {op}} \to \m... | We have:
{{begin-eqn}}
{{eqn | l = \map {F'} {g^{\text {op} } \circ f^{\text {op} } }
| r = \map {F'} {\paren{f \circ g }^{\text {op} } }
| c = {{Defof|Dual Category}}
}}
{{eqn | r = \map F {f \circ g}
| c = Definition of $F'$
}}
{{eqn | r = \map F g \circ \map F f
| c = {{Defof|Contravariant F... | Contravariant Functor Induces Covariant Functor from Dual Category | https://proofwiki.org/wiki/Contravariant_Functor_Induces_Covariant_Functor_from_Dual_Category | https://proofwiki.org/wiki/Contravariant_Functor_Induces_Covariant_Functor_from_Dual_Category | [
"Functors",
"Dual Categories"
] | [
"Definition:Metacategory",
"Definition:Functor/Contravariant",
"Definition:Dual Category",
"Definition:Object",
"Definition:Object",
"Definition:Morphism",
"Definition:Morphism",
"Definition:Functor/Covariant"
] | [
"Definition:Functor/Covariant",
"Category:Functors",
"Category:Dual Categories"
] |
proofwiki-22663 | Contravariant Functor Induces Covariant Functor to Dual Category | Let $\mathbf C$ and $\mathbf D$ be metacategories.
Let $F : \mathbf C \to \mathbf D$ be a contravariant functor.
Let $\mathbf D^{\text {op}}$ denote the dual category of $\mathbf D$.
Then $F': \mathbf C \to \mathbf D^{\text {op}}$ defined by:
:* each object $C$ of $\mathbf C$ is assigned to the object $\paren{FC}^{\tex... | We have:
{{begin-eqn}}
{{eqn | l = \map {F'} {g \circ f}
| r = \paren{\map F {g \circ f } }^{\text {op} }
| c = Definition of $F'$
}}
{{eqn | r = \paren{Ff \circ Fg}^{\text {op} }
| c = {{Defof|Contravariant Functor}}
}}
{{eqn | r = \paren{Fg}^{\text {op} } \circ \paren{Ff}^{\text {op} }
| c = {... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Metacategory|metacategories]].
Let $F : \mathbf C \to \mathbf D$ be a [[Definition:Contravariant Functor|contravariant functor]].
Let $\mathbf D^{\text {op}}$ denote the [[Definition:Dual Category|dual category]] of $\mathbf D$.
Then $F': \mathbf C \to \mathbf D^{\tex... | We have:
{{begin-eqn}}
{{eqn | l = \map {F'} {g \circ f}
| r = \paren{\map F {g \circ f } }^{\text {op} }
| c = Definition of $F'$
}}
{{eqn | r = \paren{Ff \circ Fg}^{\text {op} }
| c = {{Defof|Contravariant Functor}}
}}
{{eqn | r = \paren{Fg}^{\text {op} } \circ \paren{Ff}^{\text {op} }
| c = {... | Contravariant Functor Induces Covariant Functor to Dual Category | https://proofwiki.org/wiki/Contravariant_Functor_Induces_Covariant_Functor_to_Dual_Category | https://proofwiki.org/wiki/Contravariant_Functor_Induces_Covariant_Functor_to_Dual_Category | [
"Functors",
"Dual Categories"
] | [
"Definition:Metacategory",
"Definition:Functor/Contravariant",
"Definition:Dual Category",
"Definition:Object",
"Definition:Object",
"Definition:Morphism",
"Definition:Morphism",
"Definition:Functor/Covariant"
] | [
"Definition:Functor/Covariant",
"Category:Functors",
"Category:Dual Categories"
] |
proofwiki-22664 | Continuous Mapping Induced by Frame Homomorphism is Continuous | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be frames.
Let $f : L_1 \to L_2$ be a frame homomorphism.
Let:
:$\map {\operatorname{Sp}} f : \map {\operatorname{Sp}} {L_2} \to \map {\operatorname{Sp}} {L_1}$ denote the continuous mapping induced by $f$
where:
:considered as locales, $\map {\ope... | By definition of spectra as completely prime filters:
:$\map {\operatorname{Sp}} {L_1}$ is the topological space $\struct{\map {\operatorname{pt}} {L_1}, \set{\Sigma^{\paren 1}_{a_1} : a_1 \in L_1}}$ where:
::*$\map {\operatorname{pt}} {L_1}$ denotes the set of points as completely prime filters of $L_1$
::*$\forall a_... | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be [[Definition:Frame (Lattice Theory)|frames]].
Let $f : L_1 \to L_2$ be a [[Definition:Frame Homomorphism|frame homomorphism]].
Let:
:$\map {\operatorname{Sp}} f : \map {\operatorname{Sp}} {L_2} \to \map {\operatorname{Sp}} {L_1}$ denote the [... | By definition of [[Definition:Spectrum of Locale as Completely Prime Filters|spectra as completely prime filters]]:
:$\map {\operatorname{Sp}} {L_1}$ is the [[Definition:Topological Space|topological space]] $\struct{\map {\operatorname{pt}} {L_1}, \set{\Sigma^{\paren 1}_{a_1} : a_1 \in L_1}}$ where:
::*$\map {\operato... | Continuous Mapping Induced by Frame Homomorphism is Continuous | https://proofwiki.org/wiki/Continuous_Mapping_Induced_by_Frame_Homomorphism_is_Continuous | https://proofwiki.org/wiki/Continuous_Mapping_Induced_by_Frame_Homomorphism_is_Continuous | [
"Continuous Maps",
"Continuous Mappings"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Frame Homomorphism",
"Definition:Continuous Mapping Induced by Frame Homomorphism",
"Definition:Locale (Lattice Theory)",
"Definition:Spectrum of Locale/Completely Prime Filters",
"Definition:Continuous Mapping (Topology)/Everywhere"
] | [
"Definition:Spectrum of Locale/Completely Prime Filters",
"Definition:Topological Space",
"Definition:Set",
"Definition:Point of Locale/Completely Prime Filter",
"Definition:Topological Space",
"Definition:Set",
"Definition:Point of Locale/Completely Prime Filter",
"Definition:Continuous Mapping Induc... |
proofwiki-22665 | Brouwer's Fixed Point Theorem/Two-Dimensional Version | Let $D \subseteq \R^2$ be the closed disk defined as:
:$D = \set {\tuple {x, y} \in \R^2: x^2 + y^2 \le 1}$
Let $f: D \to D$ be a mapping which is continuous on $D$.
Then:
:$\exists \xi \in D: \map f \xi = \xi$ | {{ProofWanted}}
{{Namedfor|Luitzen Egbertus Jan Brouwer}} | Let $D \subseteq \R^2$ be the [[Definition:Closed Disk|closed disk]] defined as:
:$D = \set {\tuple {x, y} \in \R^2: x^2 + y^2 \le 1}$
Let $f: D \to D$ be a [[Definition:Mapping|mapping]] which is [[Definition:Continuous Mapping|continuous]] on $D$.
Then:
:$\exists \xi \in D: \map f \xi = \xi$ | {{ProofWanted}}
{{Namedfor|Luitzen Egbertus Jan Brouwer}} | Brouwer's Fixed Point Theorem/Two-Dimensional Version | https://proofwiki.org/wiki/Brouwer's_Fixed_Point_Theorem/Two-Dimensional_Version | https://proofwiki.org/wiki/Brouwer's_Fixed_Point_Theorem/Two-Dimensional_Version | [
"Brouwer's Fixed Point Theorem"
] | [
"Definition:Disk/Closed",
"Definition:Mapping",
"Definition:Continuous Mapping"
] | [] |
proofwiki-22666 | Injectivity of Fourier transform over Lebesgue integrable functions | Fourier transform is injective over the space $\map {\mathrm L^1} {\R^N}$
{{explain|what is $\map {\mathrm L^1} {\R^N}$?}} | Let $\FF$ denote the Fourier transform over the space of Lebesgue integrable functions $\map {\mathrm L^1} {\R^N}$.
The map $\FF$ is linear, so it is sufficient to prove that its kernel is trivial in order to show the injectivity (Linear Transformation is Injective iff Kernel Contains Only Zero).
Let $N \in \N$ and $f ... | Fourier transform is injective over the space $\map {\mathrm L^1} {\R^N}$
{{explain|what is $\map {\mathrm L^1} {\R^N}$?}} | Let $\FF$ denote the Fourier transform over the space of [[Definition:Integrable Function/Lebesgue|Lebesgue integrable functions]] $\map {\mathrm L^1} {\R^N}$.
The map $\FF$ is linear, so it is sufficient to prove that its kernel is trivial in order to show the injectivity ([[Linear Transformation is Injective iff Ker... | Injectivity of Fourier transform over Lebesgue integrable functions | https://proofwiki.org/wiki/Injectivity_of_Fourier_transform_over_Lebesgue_integrable_functions | https://proofwiki.org/wiki/Injectivity_of_Fourier_transform_over_Lebesgue_integrable_functions | [] | [] | [
"Definition:Integrable Function/Lebesgue",
"Linear Transformation is Injective iff Kernel Contains Only Zero",
"Definition:Inverse Fourier Transform/Real Function"
] |
proofwiki-22667 | Buffon's Needle/General | Let a horizontal plane be divided into strips by a series of parallel lines a fixed distance $d$ apart.
Let a needle $N$ of length $l$ such that $l < d$ be dropped onto the plane randomly from a random height.
Then the probability that $N$ across one of the parallel lines is $\dfrac {2 l} {\pi d}$. | Let the real number plane $\R^2$ divided into strips by the lines $x = k d$ for each integer $k$.
Define $\theta \in \hointr {-\dfrac \pi 2} {\dfrac \pi 2}$ as the angle between $N$ and the $x$-axis.
Then the $x$-component of the length of $N$ is $l \cos \theta$ for each $\theta$.
Let:
:$E$ be the event where $N$ falls... | Let a [[Definition:Horizontal|horizontal]] [[Definition:Plane|plane]] be divided into strips by a series of [[Definition:Parallel Lines|parallel lines]] a fixed [[Definition:Distance between Parallel Lines|distance]] $d$ apart.
Let a needle $N$ of [[Definition:Length of Line|length]] $l$ such that $l < d$ be dropped o... | Let the [[Definition:Real Number Plane|real number plane $\R^2$]] divided into strips by the lines $x = k d$ for each [[Definition:Integer|integer]] $k$.
Define $\theta \in \hointr {-\dfrac \pi 2} {\dfrac \pi 2}$ as the [[Definition:Angle|angle]] between $N$ and the [[Definition:X-Axis|$x$-axis]].
Then the [[Definit... | Buffon's Needle/General | https://proofwiki.org/wiki/Buffon's_Needle/General | https://proofwiki.org/wiki/Buffon's_Needle/General | [
"Buffon's Needle"
] | [
"Definition:Horizontal",
"Definition:Plane Surface",
"Definition:Parallel (Geometry)/Lines",
"Definition:Distance between Parallel Lines",
"Definition:Linear Measure/Length",
"Definition:Plane Surface",
"Definition:Linear Measure/Height",
"Definition:Probability",
"Definition:Parallel (Geometry)/Lin... | [
"Definition:Real Number Plane",
"Definition:Integer",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Component",
"Definition:Linear Measure/Length",
"Definition:Event",
"Definition:Vertical",
"Definition:Line/Straight Line",
"Definition:Event",
"Definition:Angle",
"Definition:Axis/X... |
proofwiki-22668 | Largest Number not Expressible as Sum of Multiples of Coprime Integers/Existence | Let $a_1,\ldots ,a_n \in \N_{>0}$.
Let $d = \gcd \set {a_1,\ldots ,a_n}$ be the greatest common divisor.
Let $d \N = \set { d k : k \in \N}$.
Let:
:$\map \N {a_1,\ldots ,a_n} = \set { k_1 a_1 + \cdots + k_n a_n : k_1,\ldots ,k_n \in \N}$
Then:
:$d \N \setminus \map \N {a_1,\ldots ,a_n}$
is a finite set.
In particular, ... | {{ProofWanted}}
Category:Largest Number not Expressible as Sum of Multiples of Coprime Integers
Category:Integer Combinations
gfdzjo6lspdspqc60se0f6a9jatnxdj | Let $a_1,\ldots ,a_n \in \N_{>0}$.
Let $d = \gcd \set {a_1,\ldots ,a_n}$ be the [[Definition:Greatest Common Divisor|greatest common divisor]].
Let $d \N = \set { d k : k \in \N}$.
Let:
:$\map \N {a_1,\ldots ,a_n} = \set { k_1 a_1 + \cdots + k_n a_n : k_1,\ldots ,k_n \in \N}$
Then:
:$d \N \setminus \map \N {a_1,\l... | {{ProofWanted}}
[[Category:Largest Number not Expressible as Sum of Multiples of Coprime Integers]]
[[Category:Integer Combinations]]
gfdzjo6lspdspqc60se0f6a9jatnxdj | Largest Number not Expressible as Sum of Multiples of Coprime Integers/Existence | https://proofwiki.org/wiki/Largest_Number_not_Expressible_as_Sum_of_Multiples_of_Coprime_Integers/Existence | https://proofwiki.org/wiki/Largest_Number_not_Expressible_as_Sum_of_Multiples_of_Coprime_Integers/Existence | [
"Largest Number not Expressible as Sum of Multiples of Coprime Integers",
"Integer Combinations"
] | [
"Definition:Greatest Common Divisor",
"Definition:Finite Set",
"Definition:Greatest Element"
] | [
"Category:Largest Number not Expressible as Sum of Multiples of Coprime Integers",
"Category:Integer Combinations"
] |
proofwiki-22669 | PDF of Cantor Distribution does not Exist | The probability density function of the Cantor distribution $C$ does not exist. | By definition, the probability density function of a probability distribution is the derivative of its cumulative distribution function.
By definition, the cumulative distribution function of $C$ is the Cantor function.
From Cantor Function is Differentiable Almost Everywhere, the Cantor function is $0$ almost everywhe... | The [[Definition:Probability Density Function|probability density function]] of the [[Definition:Cantor Distribution|Cantor distribution]] $C$ does not exist. | By definition, the [[Definition:Probability Density Function|probability density function]] of a [[Definition:Probability Distribution|probability distribution]] is the [[Definition:Derivative|derivative]] of its [[Definition:Cumulative Distribution Function|cumulative distribution function]].
By definition, the [[Def... | PDF of Cantor Distribution does not Exist | https://proofwiki.org/wiki/PDF_of_Cantor_Distribution_does_not_Exist | https://proofwiki.org/wiki/PDF_of_Cantor_Distribution_does_not_Exist | [
"Cantor Distribution",
"Probability Density Functions"
] | [
"Definition:Probability Density Function",
"Definition:Cantor Distribution"
] | [
"Definition:Probability Density Function",
"Definition:Probability Distribution",
"Definition:Derivative",
"Definition:Cumulative Distribution Function",
"Definition:Cumulative Distribution Function",
"Definition:Cantor Function",
"Cantor Function is Differentiable Almost Everywhere",
"Definition:Cant... |
proofwiki-22670 | Midpoint Characterization of Extreme Point of Convex Set | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $K$ be a convex subset of $X$.
Let $x \in K$.
Then $x$ is an extreme point of $K$ {{iff}}:
:whenever $\ds x = \frac {y + z} 2$ for $y, z \in K$, we have $x = y = z$. | === Necessary Condition ===
Suppose first that $K$ is a convex subset of $X$.
Then:
:for each $t \in \openint 0 1$, if:
::$x = t y + \paren {1 - t} z$ for some $y, z \in K$
:then $x = y = z$.
In particular, taking $t = 1/2$, it holds that:
:whenever $\ds x = \frac {y + z} 2$ for $y, z \in K$, we have $x = y = z$.
{... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $K$ be a [[Definition:Convex Set (Vector Space)|convex subset]] of $X$.
Let $x \in K$.
Then $x$ is an [[Definition:Extreme Point of Convex Set|extreme point]] of $K$ {{iff}}:
:whenever $\ds x = \frac {y + z} 2$ fo... | === Necessary Condition ===
Suppose first that $K$ is a [[Definition:Convex Set (Vector Space)|convex subset]] of $X$.
Then:
:for each $t \in \openint 0 1$, if:
::$x = t y + \paren {1 - t} z$ for some $y, z \in K$
:then $x = y = z$.
In particular, taking $t = 1/2$, it holds that:
:whenever $\ds x = \frac {y + z} ... | Midpoint Characterization of Extreme Point of Convex Set | https://proofwiki.org/wiki/Midpoint_Characterization_of_Extreme_Point_of_Convex_Set | https://proofwiki.org/wiki/Midpoint_Characterization_of_Extreme_Point_of_Convex_Set | [
"Extreme Points of Convex Sets"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Extreme Point of Convex Set"
] | [
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)"
] |
proofwiki-22671 | Characterization of Characters on Space of Complex-Valued Continuous Functions on Compact Hausdorff Space | Let $K$ be a compact Hausdorff space.
Let $\struct {\map \CC K, \overline \cdot, \norm {\, \cdot \,} }$ be the $\text C^\ast$-algebra of complex-valued continuous functions on $K$.
For each $x \in K$, define:
:$\map {\delta_x} f = \map f x$
for each $f \in \map \CC K$.
Let $\phi : \map \CC K \to \C$ be a algebra homom... | === Sufficient Condition ===
Let $x \in K$.
We show that $\delta_x$ is an algebra homomorphism.
Let $f, g \in \map \CC K$ and $\lambda \in \C$.
Then we have:
{{begin-eqn}}
{{eqn | l = \map {\delta_x} {\lambda f + g}
| r = \map {\paren {\lambda f + g} } x
}}
{{eqn | r = \map {\paren {\lambda f} } x + \map g x
}}
{{e... | Let $K$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Hausdorff Space|Hausdorff space]].
Let $\struct {\map \CC K, \overline \cdot, \norm {\, \cdot \,} }$ be the [[Definition:C*-Algebra|$\text C^\ast$-algebra]] of [[Definition:Space of Continuous Functions on Compact Hausdorff Space|complex-valued... | === Sufficient Condition ===
Let $x \in K$.
We show that $\delta_x$ is an [[Definition:Algebra Homomorphism|algebra homomorphism]].
Let $f, g \in \map \CC K$ and $\lambda \in \C$.
Then we have:
{{begin-eqn}}
{{eqn | l = \map {\delta_x} {\lambda f + g}
| r = \map {\paren {\lambda f + g} } x
}}
{{eqn | r = \map {... | Characterization of Characters on Space of Complex-Valued Continuous Functions on Compact Hausdorff Space | https://proofwiki.org/wiki/Characterization_of_Characters_on_Space_of_Complex-Valued_Continuous_Functions_on_Compact_Hausdorff_Space | https://proofwiki.org/wiki/Characterization_of_Characters_on_Space_of_Complex-Valued_Continuous_Functions_on_Compact_Hausdorff_Space | [
"Space of Continuous Functions on Compact Hausdorff Space",
"Characters (Banach Algebras)"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:C*-Algebra",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Algebra Homomorphism",
"Definition:Character (Banach Algebra)"
] | [
"Definition:Algebra Homomorphism",
"Definition:Linear Functional",
"Definition:Algebra Homomorphism",
"Constant Function is Continuous",
"Definition:Algebra Homomorphism",
"Definition:Algebra Homomorphism"
] |
proofwiki-22672 | Hermitian Element of Space of Complex-Valued Continuous Functions on Compact Hausdorff Space | Let $K$ be a compact Hausdorff space.
Let $\struct {\map \CC K, \overline \cdot, \norm {\, \cdot \,} }$ be the $\text C^\ast$-algebra of continuous functions on $K$.
Let $f \in \map \CC K$.
Then $f$ is Hermitian {{iff}} $\map f x \in \R$ for each $x \in K$. | We have that $f$ is Hermitian {{iff}} $f = \overline f$.
That is, {{iff}} $\map f x = \overline {\map f x}$ for each $x \in K$.
From Complex Number equals Conjugate iff Wholly Real, this is equivalent to $\map f x \in \R$ for each $x \in K$.
{{qed}}
Category:Space of Continuous Functions on Compact Hausdorff Space
dn... | Let $K$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Hausdorff Space|Hausdorff space]].
Let $\struct {\map \CC K, \overline \cdot, \norm {\, \cdot \,} }$ be the [[Definition:C*-Algebra|$\text C^\ast$-algebra]] of [[Definition:Space of Continuous Functions on Compact Hausdorff Space|continuous fun... | We have that $f$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]] {{iff}} $f = \overline f$.
That is, {{iff}} $\map f x = \overline {\map f x}$ for each $x \in K$.
From [[Complex Number equals Conjugate iff Wholly Real]], this is equivalent to $\map f x \in \R$ for each $x \in K$.
{{qed}}
[[Category:Spac... | Hermitian Element of Space of Complex-Valued Continuous Functions on Compact Hausdorff Space | https://proofwiki.org/wiki/Hermitian_Element_of_Space_of_Complex-Valued_Continuous_Functions_on_Compact_Hausdorff_Space | https://proofwiki.org/wiki/Hermitian_Element_of_Space_of_Complex-Valued_Continuous_Functions_on_Compact_Hausdorff_Space | [
"Space of Continuous Functions on Compact Hausdorff Space"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:C*-Algebra",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Hermitian Element of *-Algebra"
] | [
"Definition:Hermitian Element of *-Algebra",
"Complex Number equals Conjugate iff Wholly Real",
"Category:Space of Continuous Functions on Compact Hausdorff Space"
] |
proofwiki-22673 | Projection on Space of Complex-Valued Continuous Functions on Compact Hausdorff Space | Let $K$ be a compact Hausdorff space.
Let $\struct {\map \CC K, \overline \cdot, \norm {\, \cdot \,} }$ be the $\text C^\ast$-algebra of continuous functions on $K$.
Let $p \in \map \CC K$ be a projection.
Then there exists open sets (possibly empty) $U_0, U_1 \subseteq K$ such that:
:$U_0 \cap U_1 = \O$
and:
:$K = U_0... | Let $\map {B_{1/2} } 0$ and $\map {B_{1/2} } 1$ be the open balls of radius $1/2$ and centers $0$ and $1$ respectively.
Since $p$ is continuous, we have that:
:$p^{-1} \sqbrk {\map {B_{1/2} } 0}$ and $p^{-1} \sqbrk {\map {B_{1/2} } 1}$ are open in $K$.
Let $U_0 = p^{-1} \sqbrk {\map {B_{1/2} } 0}$ and $U_1 = p^{-1} \s... | Let $K$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Hausdorff Space|Hausdorff space]].
Let $\struct {\map \CC K, \overline \cdot, \norm {\, \cdot \,} }$ be the [[Definition:C*-Algebra|$\text C^\ast$-algebra]] of [[Definition:Space of Continuous Functions on Compact Hausdorff Space|continuous fun... | Let $\map {B_{1/2} } 0$ and $\map {B_{1/2} } 1$ be the [[Definition:Open Ball|open balls]] of [[Definition:Radius of Open Ball|radius]] $1/2$ and [[Definition:Center of Open Ball|centers]] $0$ and $1$ respectively.
Since $p$ is [[Definition:Continuous Mapping|continuous]], we have that:
:$p^{-1} \sqbrk {\map {B_{1/2} ... | Projection on Space of Complex-Valued Continuous Functions on Compact Hausdorff Space | https://proofwiki.org/wiki/Projection_on_Space_of_Complex-Valued_Continuous_Functions_on_Compact_Hausdorff_Space | https://proofwiki.org/wiki/Projection_on_Space_of_Complex-Valued_Continuous_Functions_on_Compact_Hausdorff_Space | [
"Space of Continuous Functions on Compact Hausdorff Space"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:C*-Algebra",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Projection (*-Algebras)",
"Definition:Open Set",
"Definition:Empty Set"
] | [
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Definition:Continuous Mapping",
"Definition:Open Set",
"Definition:Projection (*-Algebras)",
"Category:Space of Continuous Functions on Compact Hausdorff Space"
] |
proofwiki-22674 | Image of Extreme Point under Injective Linear Isomorphism is Extreme Point | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ and $Y$ be vector spaces over $\Bbb F$.
Let $K \subseteq X$ be convex.
Let $T : X \to Y$ be an injective linear transformation.
Let $x \in K$.
Then $x$ is an extreme point of $K$ {{iff}} $T x$ is an extreme point of $T \sqbrk K$. | === Necessary Condition ===
Suppose first that $x$ is an extreme point of $K$.
Let $y, z \in K$ be such that:
:$T x = t T y + \paren {1 - t} T z$
Then since $T$ is linear, we have:
:$T x = \map T {t y + \paren {1 - t} z}$
Since $T$ is injective, we have:
:$x = t y + \paren {1 - t} z$
Snce $x$ is an extreme point, we h... | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ and $Y$ be [[Definition:Vector Space|vector spaces]] over $\Bbb F$.
Let $K \subseteq X$ be [[Definition:Convex Set (Vector Space)|convex]].
Let $T : X \to Y$ be an [[Definition:Injection|injective]] [[Definition:Linear Transformation|linear transformation]].
Let $x \in K$.
... | === Necessary Condition ===
Suppose first that $x$ is an [[Definition:Extreme Point of Convex Set|extreme point]] of $K$.
Let $y, z \in K$ be such that:
:$T x = t T y + \paren {1 - t} T z$
Then since $T$ is [[Definition:Linear Transformation|linear]], we have:
:$T x = \map T {t y + \paren {1 - t} z}$
Since $T$ is ... | Image of Extreme Point under Injective Linear Isomorphism is Extreme Point | https://proofwiki.org/wiki/Image_of_Extreme_Point_under_Injective_Linear_Isomorphism_is_Extreme_Point | https://proofwiki.org/wiki/Image_of_Extreme_Point_under_Injective_Linear_Isomorphism_is_Extreme_Point | [
"Extreme Points of Convex Sets"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Injection",
"Definition:Linear Transformation",
"Definition:Extreme Point of Convex Set",
"Definition:Extreme Point of Convex Set"
] | [
"Definition:Extreme Point of Convex Set",
"Definition:Linear Transformation",
"Definition:Injection",
"Definition:Extreme Point of Convex Set",
"Definition:Extreme Point of Convex Set",
"Definition:Extreme Point of Convex Set",
"Definition:Injection",
"Definition:Extreme Point of Convex Set",
"Defin... |
proofwiki-22675 | Generalized Exponential Limit | Let $x \in \R$ be a real number.
Let $\sequence {\nu_n}$ be an increasing, unbounded above real sequence.
Let $\sequence {\xi_n}$ be a real sequence such that:
:$\ds \lim_{n \mathop \to \infty} \nu_n \xi_n = x$
Then:
:$\ds \lim_{n \mathop \to \infty} \paren {1 + \xi_n}^{\nu_n} = \map \exp x$
where $\exp$ denotes the re... | First, let us consider the case where all $\nu_n \in \Z_{\ge 0}$.
Then, by the Binomial Theorem for Integral Index:
:$\ds \paren 1 \quad \paren {1 + \xi_n}^{\nu_n} = \sum_{r \mathop = 0}^{\nu_n} \binom {\nu_n} r \xi_n^r$
Now, by Convergent Real Sequence is Bounded, there is some $X \in \R$ such that:
:$\size {\nu_j \xi... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\sequence {\nu_n}$ be an [[Definition:Increasing Real Sequence|increasing]], [[Definition:Unbounded Above Real Sequence|unbounded above real sequence]].
Let $\sequence {\xi_n}$ be a [[Definition:Real Sequence|real sequence]] such that:
:$\ds \lim_{n \ma... | First, let us consider the case where all $\nu_n \in \Z_{\ge 0}$.
Then, by the [[Binomial Theorem for Integral Index]]:
:$\ds \paren 1 \quad \paren {1 + \xi_n}^{\nu_n} = \sum_{r \mathop = 0}^{\nu_n} \binom {\nu_n} r \xi_n^r$
Now, by [[Convergent Real Sequence is Bounded]], there is some $X \in \R$ such that:
:$\size ... | Generalized Exponential Limit | https://proofwiki.org/wiki/Generalized_Exponential_Limit | https://proofwiki.org/wiki/Generalized_Exponential_Limit | [
"Exponential Function"
] | [
"Definition:Real Number",
"Definition:Increasing/Sequence/Real Sequence",
"Definition:Bounded Above Sequence/Real/Unbounded",
"Definition:Real Sequence",
"Definition:Exponential Function/Real",
"Definition:Exponential Function/Real/Power Series Expansion"
] | [
"Binomial Theorem/Integral Index",
"Convergent Real Sequence is Bounded",
"Tannery's Theorem",
"Binomial Coefficient over Power Not Greater than Reciprocal of Factorial",
"Radius of Convergence of Power Series over Factorial",
"Definition:Convergent Series/Number Field",
"Combination Theorem for Sequenc... |
proofwiki-22676 | Intersection of Closed Subsets satisfying Finite Intersection Property in Compact Space | Let $T = \struct {S, \tau}$ be a compact topological space.
Let $\FF = \set {V_\alpha}_{\alpha \mathop \in I} \subseteq \powerset S$ be a family of closed sets in $T$.
Suppose that $\FF$ satisfies the finite intersection property.
:That is, the intersection of every finite subset is non-empty.
Then the intersection of ... | {{AimForCont}} $\ds \bigcap_{\alpha \mathop \in I} V_\alpha = \O$.
Then, by De Morgan's Laws:
:$\ds \bigcup_{\alpha \mathop \in I} \relcomp S {V_\alpha} = \relcomp S \O = S$
Since each $V_\alpha$ is closed, $\relcomp S {V_\alpha}$ is open.
Hence, $\set {\relcomp S {V_\alpha}}_{\alpha \mathop \in I}$ forms an open cover... | Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $\FF = \set {V_\alpha}_{\alpha \mathop \in I} \subseteq \powerset S$ be a [[Definition:Indexed Family of Sets|family]] of [[Definition:Closed Set (Topology)|closed sets]] in $T$.
Suppose that $\FF$ satisfies the [... | {{AimForCont}} $\ds \bigcap_{\alpha \mathop \in I} V_\alpha = \O$.
Then, by [[De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection|De Morgan's Laws]]:
:$\ds \bigcup_{\alpha \mathop \in I} \relcomp S {V_\alpha} = \relcomp S \O = S$
Since each $V_\alpha$ is [[Definition:Closed Set (Top... | Intersection of Closed Subsets satisfying Finite Intersection Property in Compact Space | https://proofwiki.org/wiki/Intersection_of_Closed_Subsets_satisfying_Finite_Intersection_Property_in_Compact_Space | https://proofwiki.org/wiki/Intersection_of_Closed_Subsets_satisfying_Finite_Intersection_Property_in_Compact_Space | [
"Compact Topological Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:Indexing Set/Family of Sets",
"Definition:Closed Set/Topology",
"Definition:Finite Intersection Property",
"Definition:Set Intersection/Family of Sets",
"Definition:Finite Subset",
"Definition:Non-Empty Set",
"Definition:Set Intersection/Family of Se... | [
"De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Cover",
"Definition:Compact Topological Space",
"Definition:Subcover/Finite",
"De Morgan's Laws (Set Theory)/Set Complement/Family ... |
proofwiki-22677 | Number of Spanning Trees on Complete Graph | Let $K_n$ denote the complete graph of order $n$.
The number of spanning trees on $K_n$ is equal to $n^{n - 2}$. | {{ProofWanted|Induction may work here}} | Let $K_n$ denote the [[Definition:Complete Graph|complete graph]] of [[Definition:Order of Graph|order]] $n$.
The number of [[Definition:Spanning Tree|spanning trees]] on $K_n$ is equal to $n^{n - 2}$. | {{ProofWanted|Induction may work here}} | Number of Spanning Trees on Complete Graph | https://proofwiki.org/wiki/Number_of_Spanning_Trees_on_Complete_Graph | https://proofwiki.org/wiki/Number_of_Spanning_Trees_on_Complete_Graph | [
"Complete Graphs",
"Spanning Trees"
] | [
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Spanning Tree"
] | [] |
proofwiki-22678 | Closed Unit Ball of Space of Zero-Limit Sequences has no Extreme Points | Let $\struct {c_0, \norm {\, \cdot \,}_\infty}$ be the Banach space of zero-limit sequences.
Let $B_{c_0}^-$ be the closed unit ball of $\struct {c_0, \norm {\, \cdot \,}_\infty}$.
Then $B_{c_0}^-$ has no extreme points. | From Space of Zero-Limit Sequences with Supremum Norm forms Banach Space, $\struct {c_0, \norm {\, \cdot \,}_\infty}$ is a Banach space.
From Closed Unit Ball is Convex Set, $B_{c_0}^-$ is convex and hence we can speak of its extreme points..
Let $\sequence {x_n}_{n \mathop \in \N} \in B_{c_0}^-$.
Since $x_n \to 0$, t... | Let $\struct {c_0, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Banach Space|Banach space]] of [[Definition:Space of Zero-Limit Sequences|zero-limit sequences]].
Let $B_{c_0}^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] of $\struct {c_0, \norm {\, \cdot \,}_\infty}$.
Then $B_{c_0}^-$ has no [[Defin... | From [[Space of Zero-Limit Sequences with Supremum Norm forms Banach Space]], $\struct {c_0, \norm {\, \cdot \,}_\infty}$ is a [[Definition:Banach Space|Banach space]].
From [[Closed Unit Ball is Convex Set]], $B_{c_0}^-$ is [[Definition:Convex Set (Vector Space)|convex]] and hence we can speak of its [[Definition:Ext... | Closed Unit Ball of Space of Zero-Limit Sequences has no Extreme Points | https://proofwiki.org/wiki/Closed_Unit_Ball_of_Space_of_Zero-Limit_Sequences_has_no_Extreme_Points | https://proofwiki.org/wiki/Closed_Unit_Ball_of_Space_of_Zero-Limit_Sequences_has_no_Extreme_Points | [
"Extreme Points of Convex Sets"
] | [
"Definition:Banach Space",
"Definition:Space of Zero-Limit Sequences",
"Definition:Closed Unit Ball",
"Definition:Extreme Point of Convex Set"
] | [
"Space of Zero-Limit Sequences with Supremum Norm forms Banach Space",
"Definition:Banach Space",
"Closed Unit Ball is Convex Set",
"Definition:Convex Set (Vector Space)",
"Definition:Extreme Point of Convex Set",
"Definition:Vector Space",
"Triangle Inequality/Complex Numbers",
"Definition:Extreme Po... |
proofwiki-22679 | Ceiling Function/Examples/Ceiling of 3.2 | :$\ceiling {3 \cdotp 2} = 4$ | We have that:
:$3 < 3 \cdotp 2 \le 4$
Hence $4$ is the ceiling of $3 \cdotp 2$ by definition.
{{qed}} | :$\ceiling {3 \cdotp 2} = 4$ | We have that:
:$3 < 3 \cdotp 2 \le 4$
Hence $4$ is the [[Definition:Ceiling Function|ceiling]] of $3 \cdotp 2$ by definition.
{{qed}} | Ceiling Function/Examples/Ceiling of 3.2 | https://proofwiki.org/wiki/Ceiling_Function/Examples/Ceiling_of_3.2 | https://proofwiki.org/wiki/Ceiling_Function/Examples/Ceiling_of_3.2 | [
"Examples of Ceiling Function"
] | [] | [
"Definition:Ceiling Function"
] |
proofwiki-22680 | Ceiling Function/Examples/Ceiling of 5 | :$\ceiling 5 = 5$ | We have that:
:$4 < 5 \le 5$
Hence $5$ is the ceiling of $5$ by definition.
{{qed}} | :$\ceiling 5 = 5$ | We have that:
:$4 < 5 \le 5$
Hence $5$ is the [[Definition:Ceiling Function|ceiling]] of $5$ by definition.
{{qed}} | Ceiling Function/Examples/Ceiling of 5 | https://proofwiki.org/wiki/Ceiling_Function/Examples/Ceiling_of_5 | https://proofwiki.org/wiki/Ceiling_Function/Examples/Ceiling_of_5 | [
"Examples of Ceiling Function"
] | [] | [
"Definition:Ceiling Function"
] |
proofwiki-22681 | Intersections of Slices form Neighborhood Basis for Standard Topology of Compact Convex Subset of Hausdorff Locally Convex Space | Let $\struct {X, \PP}$ be a Hausdorff locally convex space with its standard topology.
Let $X^\ast$ be the topological dual space of $X$.
Let $K$ be a compact convex subset of $X$.
Let $x_0 \in K$.
Let $V$ be a open neighborhood of $x_0$ in $K$.
Let $\FF$ be the set of non-empty finite subsets of $X^\ast$.
Let:
:$V_{... | Since $V$ is open, $K \setminus V$ is closed.
Let $x \in K \setminus V$.
Since $\struct {X, \PP}$ is Hausdorff, $\set {x_0}$ is compact.
From:
:Hahn-Banach Separation Theorem: Hausdorff Locally Convex Space: Compact Convex Set and Closed Convex Set (Real Case) if $\GF = \R$
:Hahn-Banach Separation Theorem: Hausdorff Lo... | Let $\struct {X, \PP}$ be a [[Definition:Hausdorff Locally Convex Space|Hausdorff locally convex space]] with its [[Definition:Locally Convex Space/Standard Topology|standard topology]].
Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual space]] of $X$.
Let $K$ be a [[Definition:Compact Topologi... | Since $V$ is [[Definition:Open Set|open]], $K \setminus V$ is [[Definition:Closed Set|closed]].
Let $x \in K \setminus V$.
Since $\struct {X, \PP}$ is [[Definition:Hausdorff Space|Hausdorff]], $\set {x_0}$ is [[Definition:Compact Topological Space|compact]].
From:
:[[Hahn-Banach Separation Theorem/Hausdorff Locally ... | Intersections of Slices form Neighborhood Basis for Standard Topology of Compact Convex Subset of Hausdorff Locally Convex Space | https://proofwiki.org/wiki/Intersections_of_Slices_form_Neighborhood_Basis_for_Standard_Topology_of_Compact_Convex_Subset_of_Hausdorff_Locally_Convex_Space | https://proofwiki.org/wiki/Intersections_of_Slices_form_Neighborhood_Basis_for_Standard_Topology_of_Compact_Convex_Subset_of_Hausdorff_Locally_Convex_Space | [
"Locally Convex Spaces",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Locally Convex Space/Hausdorff",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Topological Dual Space",
"Definition:Compact Topological Space",
"Definition:Convex Set (Vector Space)",
"Definition:Open Neighborhood",
"Definition:Set",
"Definition:Non-Empty Set",
"Defini... | [
"Definition:Open Set",
"Definition:Closed Set",
"Definition:T2 Space",
"Definition:Compact Topological Space",
"Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Real Case/Compact Convex Set and Closed Convex Set",
"Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Complex Case/Com... |
proofwiki-22682 | Slices form Neighborhood Basis of Extreme Point of Compact Convex Subset of Hausdorff Locally Convex Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\GF$ with its standard topology.
Let $X^\ast$ be the topological dual space of $X$.
Let $K$ be a compact convex subset of $X$.
Let $x_0$ be an extreme point of $K$.
Let $V$ be a open neighborhood of $x_0$ in $K$.
Let:
:$V_{f,... | From Intersections of Slices form Neighborhood Basis for Standard Topology of Compact Convex Subset of Hausdorff Locally Convex Space, there exists $f_1, \ldots, f_n \in X^\ast$ and $\alpha_1, \ldots, \alpha_n \in \R$ such that:
:$\ds x_0 \in \bigcap_{i \mathop = 1}^n \set {x \in K : \map \Re {\map {f_i} x} < \alpha_i}... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Hausdorff Locally Convex Space|Hausdorff locally convex space]] over $\GF$ with its [[Definition:Locally Convex Space/Standard Topology|standard topology]].
Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual space]] of $X$.
L... | From [[Intersections of Slices form Neighborhood Basis for Standard Topology of Compact Convex Subset of Hausdorff Locally Convex Space]], there exists $f_1, \ldots, f_n \in X^\ast$ and $\alpha_1, \ldots, \alpha_n \in \R$ such that:
:$\ds x_0 \in \bigcap_{i \mathop = 1}^n \set {x \in K : \map \Re {\map {f_i} x} < \alph... | Slices form Neighborhood Basis of Extreme Point of Compact Convex Subset of Hausdorff Locally Convex Space | https://proofwiki.org/wiki/Slices_form_Neighborhood_Basis_of_Extreme_Point_of_Compact_Convex_Subset_of_Hausdorff_Locally_Convex_Space | https://proofwiki.org/wiki/Slices_form_Neighborhood_Basis_of_Extreme_Point_of_Compact_Convex_Subset_of_Hausdorff_Locally_Convex_Space | [
"Locally Convex Spaces",
"Extreme Points of Convex Sets"
] | [
"Definition:Locally Convex Space/Hausdorff",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Topological Dual Space",
"Definition:Compact Topological Space",
"Definition:Convex Set (Vector Space)",
"Definition:Extreme Point of Convex Set",
"Definition:Open Neighborhood"
] | [
"Intersections of Slices form Neighborhood Basis for Standard Topology of Compact Convex Subset of Hausdorff Locally Convex Space",
"Definition:Continuous Mapping",
"Real Part of Continuous Function is Continuous",
"Definition:Closed Set",
"Closed Subspace of Compact Space is Compact",
"Definition:Compact... |
proofwiki-22683 | Convex Combination contained in Convex Set | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $K$ be a convex subset of $X$.
Let $\lambda_1, \ldots, \lambda_n \in \openint 0 1$ be such that:
:$\ds \sum_{i \mathop = 1}^n \lambda_i = 1$
Let $u_1, \ldots, u_n \in K$.
Then we have:
:$\ds \sum_{i \mathop = 1}^n \lambda_i u_i \in K$ | For each $n \in \N$, let $\map P n$ be the proposition:
:for each $\lambda_1, \lambda_2, \ldots, \lambda_n \in \openint 0 1$ with:
::$\ds \sum_{i \mathop = 1}^n \lambda_i = 1$
:and $u_1, u_2, \ldots, u_n \in U$, we have:
::$\ds \sum_{i \mathop = 1}^n \lambda_i u_i \in K$
We proceed by induction. | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $K$ be a [[Definition:Convex Set (Vector Space)|convex subset]] of $X$.
Let $\lambda_1, \ldots, \lambda_n \in \openint 0 1$ be such that:
:$\ds \sum_{i \mathop = 1}^n \lambda_i = 1$
Let $u_1, \ldots, u_n \in K$.
Th... | For each $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:for each $\lambda_1, \lambda_2, \ldots, \lambda_n \in \openint 0 1$ with:
::$\ds \sum_{i \mathop = 1}^n \lambda_i = 1$
:and $u_1, u_2, \ldots, u_n \in U$, we have:
::$\ds \sum_{i \mathop = 1}^n \lambda_i u_i \in K$
We proceed by [[Pr... | Convex Combination contained in Convex Set | https://proofwiki.org/wiki/Convex_Combination_contained_in_Convex_Set | https://proofwiki.org/wiki/Convex_Combination_contained_in_Convex_Set | [
"Convex Sets (Vector Spaces)"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-22684 | Extreme Point of Convex Set not contained in Convex Hull of Subset not containing Point | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $K$ be a convex subset of $X$.
Let $x \in K$ be an extreme point of $K$.
Let $C \subseteq K$ be such that $x \not \in C$.
Let $\map {\operatorname {conv} } C$ be the convex hull of $C$.
Then $x \not \in \map {\operatorname {conv} } C$. | Suppose that $x \in \map {\operatorname {conv} } C$.
Let $\lambda_0, \ldots, \lambda_n \in \openint 0 1$ and $u_0, \ldots, u_n \in C$ be such that:
:$\ds \sum_{i \mathop = 0}^n \lambda_i = 1$
and:
:$\ds x = \sum_{i \mathop = 0}^n \lambda_i u_i$
Write:
:$\ds x = \lambda_0 u_0 + \sum_{i \mathop = 1}^n \lambda_i u_i$
and ... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $K$ be a [[Definition:Convex Set (Vector Space)|convex subset]] of $X$.
Let $x \in K$ be an [[Definition:Extreme Point of Convex Set|extreme point]] of $K$.
Let $C \subseteq K$ be such that $x \not \in C$.
Let $\m... | Suppose that $x \in \map {\operatorname {conv} } C$.
Let $\lambda_0, \ldots, \lambda_n \in \openint 0 1$ and $u_0, \ldots, u_n \in C$ be such that:
:$\ds \sum_{i \mathop = 0}^n \lambda_i = 1$
and:
:$\ds x = \sum_{i \mathop = 0}^n \lambda_i u_i$
Write:
:$\ds x = \lambda_0 u_0 + \sum_{i \mathop = 1}^n \lambda_i u_i$
an... | Extreme Point of Convex Set not contained in Convex Hull of Subset not containing Point | https://proofwiki.org/wiki/Extreme_Point_of_Convex_Set_not_contained_in_Convex_Hull_of_Subset_not_containing_Point | https://proofwiki.org/wiki/Extreme_Point_of_Convex_Set_not_contained_in_Convex_Hull_of_Subset_not_containing_Point | [
"Convex Hulls",
"Extreme Points of Convex Sets"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Extreme Point of Convex Set",
"Definition:Convex Hull"
] | [
"Convex Combination contained in Convex Set",
"Definition:Extreme Point of Convex Set",
"Category:Convex Hulls",
"Category:Extreme Points of Convex Sets"
] |
proofwiki-22685 | Convex Hull of Finite Union of Convex Sets | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $K_1, \ldots, K_n \subseteq X$ be convex.
Let:
:$\ds K = \bigcup_{i \mathop = 1}^n K_i$
Let $\map {\operatorname {conv} } K$ be the convex hull of $K$.
Then:
:$\ds \map {\operatorname {conv} } K = \set {\sum_{i \mathop = 1}^n \lambda_i x_i : \lambd... | We first show that:
:$\ds \set {\sum_{i \mathop = 1}^n \lambda_i x_i : \lambda_i \in \R_{> 0}, \, x_i \in K_i, \, \sum_{i \mathop = 1}^n t_i = 1} \subseteq \map {\operatorname {conv} } K$
Let $x_i \in K_i$ for $i \in \set {1, \ldots, n}$ and $t_1, \ldots, t_n \in \R_{> 0}$ be such that:
:$\ds \sum_{i \mathop = 1}^n t_i... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $K_1, \ldots, K_n \subseteq X$ be [[Definition:Convex Set (Vector Space)|convex]].
Let:
:$\ds K = \bigcup_{i \mathop = 1}^n K_i$
Let $\map {\operatorname {conv} } K$ be the [[Definition:Convex Hull|convex hull]] of $... | We first show that:
:$\ds \set {\sum_{i \mathop = 1}^n \lambda_i x_i : \lambda_i \in \R_{> 0}, \, x_i \in K_i, \, \sum_{i \mathop = 1}^n t_i = 1} \subseteq \map {\operatorname {conv} } K$
Let $x_i \in K_i$ for $i \in \set {1, \ldots, n}$ and $t_1, \ldots, t_n \in \R_{> 0}$ be such that:
:$\ds \sum_{i \mathop = 1}^n t_... | Convex Hull of Finite Union of Convex Sets | https://proofwiki.org/wiki/Convex_Hull_of_Finite_Union_of_Convex_Sets | https://proofwiki.org/wiki/Convex_Hull_of_Finite_Union_of_Convex_Sets | [
"Convex Sets (Vector Spaces)",
"Convex Hulls"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Hull"
] | [
"Convex Combination contained in Convex Set",
"Definition:Pairwise Disjoint",
"Definition:Convex Set (Vector Space)",
"Convex Combination contained in Convex Set",
"Category:Convex Sets (Vector Spaces)",
"Category:Convex Hulls"
] |
proofwiki-22686 | Closure of Set with Compact Closed Convex Hull in Hausdorff Locally Convex Space contains Extreme Points of Convex Hull | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\GF$ with its standard topology.
Let $S \subseteq X$ be such that:
:$K = \map \cl {\map {\operatorname {conv} } S}$ is compact
where $\operatorname {conv}$ denotes convex hull.
Let $\map E K$ be the set of extreme points of $K... | Let $X^\ast$ be the topological dual space of $X$.
{{AimForCont}} suppose that:
:$\map E K \not \subseteq \map \cl S$
Take $x_0 \in \map E K \setminus \map \cl S$.
Then $X \setminus \map \cl S$ is an open neighborhood of $x_0$ in $X$.
By Slices form Neighborhood Basis of Extreme Point of Compact Convex Subset of Hausdo... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Hausdorff Locally Convex Space|Hausdorff locally convex space]] over $\GF$ with its [[Definition:Locally Convex Space/Standard Topology|standard topology]].
Let $S \subseteq X$ be such that:
:$K = \map \cl {\map {\operatorname {conv} } S}$ is [[Def... | Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual space]] of $X$.
{{AimForCont}} suppose that:
:$\map E K \not \subseteq \map \cl S$
Take $x_0 \in \map E K \setminus \map \cl S$.
Then $X \setminus \map \cl S$ is an [[Definition:Open Neighborhood|open neighborhood]] of $x_0$ in $X$.
By [[Slice... | Closure of Set with Compact Closed Convex Hull in Hausdorff Locally Convex Space contains Extreme Points of Convex Hull | https://proofwiki.org/wiki/Closure_of_Set_with_Compact_Closed_Convex_Hull_in_Hausdorff_Locally_Convex_Space_contains_Extreme_Points_of_Convex_Hull | https://proofwiki.org/wiki/Closure_of_Set_with_Compact_Closed_Convex_Hull_in_Hausdorff_Locally_Convex_Space_contains_Extreme_Points_of_Convex_Hull | [
"Convex Hulls",
"Extreme Points of Convex Sets",
"Locally Convex Spaces"
] | [
"Definition:Locally Convex Space/Hausdorff",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Compact Topological Space",
"Definition:Convex Hull",
"Definition:Set",
"Definition:Extreme Point of Convex Set"
] | [
"Definition:Topological Dual Space",
"Definition:Open Neighborhood",
"Slices form Neighborhood Basis of Extreme Point of Compact Convex Subset of Hausdorff Locally Convex Space",
"Definition:Continuous Mapping",
"Definition:Continuous Mapping",
"Real Part of Continuous Function is Continuous",
"Definiti... |
proofwiki-22687 | Product of Cardinalities of Centralizer and Conjugacy Class | Let $\struct {G, \circ}$ be a group and let $a \in G$.
Let $\map {C_G} a$ denote the centralizer of $a$ in $G$.
Let $\conjclass a$ denote the conjugacy class of $a$ in $G$.
Then:
:$\order {\map {C_G} a} \times \order {\conjclass a} = \order G$
where $\order G$ denotes the order of $G$. | From Number of Conjugates is Number of Cosets of Centralizer:
:$\order {\conjclass a} = \index G {\map {C_G} a}$
where $\index G {\map {C_G} a}$ is the index of $\map {C_G} a$ in $G$.
From Lagrange's Theorem (Group Theory):
:$\index G {\map {C_G} a} = \dfrac {\order G} {\order {\map {C_G} a} }$
Hence the result:
:$\ord... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] and let $a \in G$.
Let $\map {C_G} a$ denote the [[Definition:Centralizer of Group Element|centralizer]] of $a$ in $G$.
Let $\conjclass a$ denote the [[Definition:Conjugacy Class|conjugacy class]] of $a$ in $G$.
Then:
:$\order {\map {C_G} a} \times \order {\co... | From [[Number of Conjugates is Number of Cosets of Centralizer]]:
:$\order {\conjclass a} = \index G {\map {C_G} a}$
where $\index G {\map {C_G} a}$ is the [[Definition:Index of Subgroup|index]] of $\map {C_G} a$ in $G$.
From [[Lagrange's Theorem (Group Theory)]]:
:$\index G {\map {C_G} a} = \dfrac {\order G} {\order... | Product of Cardinalities of Centralizer and Conjugacy Class | https://proofwiki.org/wiki/Product_of_Cardinalities_of_Centralizer_and_Conjugacy_Class | https://proofwiki.org/wiki/Product_of_Cardinalities_of_Centralizer_and_Conjugacy_Class | [
"Centralizers",
"Conjugacy Classes"
] | [
"Definition:Group",
"Definition:Centralizer/Group Element",
"Definition:Conjugacy Class",
"Definition:Order of Structure"
] | [
"Number of Conjugates is Number of Cosets of Centralizer",
"Definition:Index of Subgroup",
"Lagrange's Theorem (Group Theory)"
] |
proofwiki-22688 | Dilworth's Theorem | Let $\struct {S, \preccurlyeq}$ be a finite ordered set.
Then the maximal size of an antichain is equal to the minimal number of pairwise disjoint chains in $\struct {S, \preccurlyeq}$ that partition $S$.
That is:
:$\min \set {k : \exists C_1, \dots C_k \subseteq S \text { chains such that } C_1 \mathop {\dot \cup} \do... | Let $m$ denote the minimal number of chains needed to partition $S$.
Let $M$ denote the maximal size of an antichain in $S$.
Moreover let $A$ be an antichain of size $M$.
Let $C_1, C_2 \dots C_m$ be chains that partition $S$.
By definition of partition:
:$C_1, C_2 \dots C_m$ are pairwise disjoint
:$C_1, C_2 \dots C_m$ ... | Let $\struct {S, \preccurlyeq}$ be a [[Definition:Finite Set|finite]] [[Definition:Ordered Set|ordered set]].
Then the maximal size of an [[Definition:Antichain|antichain]] is equal to the minimal number of [[Definition:Pairwise Disjoint|pairwise disjoint]] [[Definition:Chain (Order Theory)|chains]] in $\struct {S, \p... | Let $m$ denote the minimal number of [[Definition:Chain (Order Theory)|chains]] needed to [[Definition:Set Partition|partition]] $S$.
Let $M$ denote the maximal size of an [[Definition:Antichain|antichain]] in $S$.
Moreover let $A$ be an [[Definition:Antichain|antichain]] of size $M$.
Let $C_1, C_2 \dots C_m$ be [[D... | Dilworth's Theorem | https://proofwiki.org/wiki/Dilworth's_Theorem | https://proofwiki.org/wiki/Dilworth's_Theorem | [
"Order Theory",
"Graph Theory",
"Antichains"
] | [
"Definition:Finite Set",
"Definition:Ordered Set",
"Definition:Antichain",
"Definition:Pairwise Disjoint",
"Definition:Chain (Order Theory)",
"Definition:Set Partition",
"Definition:Disjoint Union"
] | [
"Definition:Chain (Order Theory)",
"Definition:Set Partition",
"Definition:Antichain",
"Definition:Antichain",
"Definition:Chain (Order Theory)",
"Definition:Set Partition",
"Definition:Set Partition",
"Definition:Pairwise Disjoint",
"Definition:Cover of Set",
"Definition:Antichain",
"Definition... |
proofwiki-22689 | Degree of Characteristic Polynomial of Matrix equals Order of Matrix | Let $R$ be a commutative ring with unity.
Let $\mathbf A$ be a square matrix over $R$ of order $n > 0$.
Let $\mathbf I_n$ be the $n \times n$ identity matrix.
Let $R \sqbrk x$ be the polynomial ring in one variable over $R$.
Let $\map {p_{\mathbf A} } x$ be the characteristic polynomial of $\mathbf A$.
Then the degree ... | From Existence of Schur Decomposition for Square Matrix, matrix $\mathbf A$ is similar to an upper triangular matrix $\mathbf T$.
Hence:
{{begin-eqn}}
{{eqn | l = \map {p_{\mathbf A} } x
| r = \map \det {x \mathbf I_n - \mathbf A}
| c = {{Defof|Characteristic Polynomial of Matrix}} (alternate definition)
}}... | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] over $R$ of [[Definition:Order of Square Matrix|order]] $n > 0$.
Let $\mathbf I_n$ be the $n \times n$ [[Definition:Identity Matrix|identity matrix]].
Let $R \sqbrk x$ ... | From [[Existence of Schur Decomposition for Square Matrix]], [[Definition:Matrix|matrix]] $\mathbf A$ is [[Definition:Matrix Similarity|similar]] to an [[Definition:Upper Triangular Matrix|upper triangular matrix]] $\mathbf T$.
Hence:
{{begin-eqn}}
{{eqn | l = \map {p_{\mathbf A} } x
| r = \map \det {x \mathbf I... | Degree of Characteristic Polynomial of Matrix equals Order of Matrix | https://proofwiki.org/wiki/Degree_of_Characteristic_Polynomial_of_Matrix_equals_Order_of_Matrix | https://proofwiki.org/wiki/Degree_of_Characteristic_Polynomial_of_Matrix_equals_Order_of_Matrix | [
"Characteristic Polynomial of Matrix"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Unit Matrix",
"Definition:Polynomial Ring",
"Definition:Characteristic Polynomial of Matrix",
"Definition:Degree of Polynomial",
"Definition:Matrix/Square Matrix/Order"
] | [
"Existence of Schur Decomposition for Square Matrix",
"Definition:Matrix",
"Definition:Matrix Similarity",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Similar Matrices have Same Characteristic Polynomial",
"Determinant of Upper Triangular Matrix",
"Definition:Main Diagonal/Diagonal Elements... |
proofwiki-22690 | Character is Constant on Conjugacy Class | Let $G$ be a finite group.
Let $\conjclass g$ be the conjugacy class for some $g \in G$.
Let $\chi_\rho$ be the character associated with an irreducible linear representation $\rho$ of $G$.
Then:
:$\forall x \in \conjclass g: \map {\chi_\rho} x = \map {\chi_\rho} g$
That is, character is constant across all elements of... | {{Recall|Character (Representation Theory)|character}}
{{:Definition:Character (Representation Theory)}}
Let $x = h^{-1} g h$ for some $h \in G$.
{{begin-eqn}}
{{eqn | l = \map {\chi_\rho} x
| r = \map \tr {\map \rho x}
}}
{{eqn | r = \map \tr {\map \rho {h^{-1} g h} }
}}
{{eqn | r = \map \tr {\paren {\map \rho h... | Let $G$ be a [[Definition:Finite Group|finite group]].
Let $\conjclass g$ be the [[Definition:Conjugacy Class|conjugacy class]] for some $g \in G$.
Let $\chi_\rho$ be the [[Definition:Character (Representation Theory)|character]] associated with an [[Definition:Irreducible Linear Representation|irreducible]] [[Defini... | {{Recall|Character (Representation Theory)|character}}
{{:Definition:Character (Representation Theory)}}
Let $x = h^{-1} g h$ for some $h \in G$.
{{begin-eqn}}
{{eqn | l = \map {\chi_\rho} x
| r = \map \tr {\map \rho x}
}}
{{eqn | r = \map \tr {\map \rho {h^{-1} g h} }
}}
{{eqn | r = \map \tr {\paren {\map \rho... | Character is Constant on Conjugacy Class | https://proofwiki.org/wiki/Character_is_Constant_on_Conjugacy_Class | https://proofwiki.org/wiki/Character_is_Constant_on_Conjugacy_Class | [
"Conjugacy Classes",
"Character (Representation Theory)"
] | [
"Definition:Finite Group",
"Definition:Conjugacy Class",
"Definition:Character (Representation Theory)",
"Definition:Irreducible (Representation Theory)/Linear Representation",
"Definition:Linear Representation/Group",
"Definition:Character (Representation Theory)",
"Definition:Element",
"Definition:C... | [
"Definition:Group Homomorphism",
"Similar Matrices have same Traces"
] |
proofwiki-22691 | Zero Sets are Exactly the Closed G-Delta Sets in T4 Space | Let $T = \struct {S, \tau}$ be a $T_4$ topological space.
Then the zero sets in $T$ are precisely the closed $G_\delta$ subsets of $S$. | Let $T = \struct {S, \tau}$ be a $T_4$ space. | Let $T = \struct {S, \tau}$ be a [[Definition:T4 Space|$T_4$ topological space]].
Then the [[Definition:Zero Set|zero sets]] in $T$ are precisely the [[Definition:Closed Set (Topology)|closed]] [[Definition:G-Delta Set|$G_\delta$ subsets]] of $S$. | Let $T = \struct {S, \tau}$ be a [[Definition:T4 Space|$T_4$ space]]. | Zero Sets are Exactly the Closed G-Delta Sets in T4 Space | https://proofwiki.org/wiki/Zero_Sets_are_Exactly_the_Closed_G-Delta_Sets_in_T4_Space | https://proofwiki.org/wiki/Zero_Sets_are_Exactly_the_Closed_G-Delta_Sets_in_T4_Space | [
"T4 Spaces",
"Zero Sets",
"Closed Sets",
"G-Delta Sets"
] | [
"Definition:T4 Space",
"Definition:Zero Set",
"Definition:Closed Set/Topology",
"Definition:G-Delta Set"
] | [
"Definition:T4 Space"
] |
proofwiki-22692 | Coordinate Functionals Associated with Schauder Basis of Banach Space are Bounded | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\GF$.
Let $E = \sequence {e_n}_{n \mathop \in \N}$ be a Schauder basis for $X$.
Let $\sequence {e_n^\ast}_{n \mathop \in \N}$ be the coordinate functionals associated with $\sequence {x_n}_{n \mathop \in \N}$.
Then each linear... | We define the projections associated with $\sequence {e_n}_{n \mathop \in \N}$ by:
:$\ds \map {S_n} x = \sum_{k \mathop = 1}^n \map {e_k^\ast} x e_k$
Define $\norm {\, \cdot \,}_E : X \to \R$ by:
:$\ds \norm x_E = \sup_{n \mathop \in \N} \norm {\map {S_n} x}$ for each $x \in X$.
From Norm Induced by Projections Associ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $E = \sequence {e_n}_{n \mathop \in \N}$ be a [[Definition:Schauder Basis|Schauder basis]] for $X$.
Let $\sequence {e_n^\ast}_{n \mathop \in \N}$ be the [[Definition:Coordinate Functiona... | We define the [[Definition:Projections Associated with Schauder Basis|projections associated with $\sequence {e_n}_{n \mathop \in \N}$]] by:
:$\ds \map {S_n} x = \sum_{k \mathop = 1}^n \map {e_k^\ast} x e_k$
Define $\norm {\, \cdot \,}_E : X \to \R$ by:
:$\ds \norm x_E = \sup_{n \mathop \in \N} \norm {\map {S_n} x}$ f... | Coordinate Functionals Associated with Schauder Basis of Banach Space are Bounded | https://proofwiki.org/wiki/Coordinate_Functionals_Associated_with_Schauder_Basis_of_Banach_Space_are_Bounded | https://proofwiki.org/wiki/Coordinate_Functionals_Associated_with_Schauder_Basis_of_Banach_Space_are_Bounded | [
"Coordinate Functionals Associated with Schauder Bases",
"Coordinate Functionals Associated with Schauder Basis of Banach Space are Bounded"
] | [
"Definition:Banach Space",
"Definition:Schauder Basis",
"Definition:Coordinate Functionals Associated with Schauder Basis",
"Definition:Linear Functional",
"Definition:Bounded Linear Functional"
] | [
"Definition:Projections Associated with Schauder Basis",
"Norm Induced by Projections Associated with Schauder Basis is Norm",
"Definition:Norm/Vector Space",
"Definition:Identity Mapping",
"Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology",
"Characterization of ... |
proofwiki-22693 | Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology | Let $\GF \in \set {\R, \C}$.
Let $X$ be a normed vector space over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a Schauder basis for $X$.
Let $\sequence {S_n}_{n \mathop \in \N}$ be the projections associated with $\sequence {x_n}_{n \mathop \in \N}$.
Then $\sequence {S_n}_{n \mathop \in \N}$ converges to the id... | From Characterization of Convergence in Strong Operator Topology, it suffices to show:
:$S_n x \to x$ for each $x \in X$.
Let $\sequence {x_n^\ast}_{n \mathop \in \N}$ be the coordinate functionals associated with $\sequence {x_n}_{n \mathop \in \N}$.
By the definition of a Schauder basis, we have:
:$\ds x = \sum_{k \m... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Schauder Basis|Schauder basis]] for $X$.
Let $\sequence {S_n}_{n \mathop \in \N}$ be the [[Definition:Projections Associated with Schauder Basis|proj... | From [[Characterization of Convergence in Strong Operator Topology]], it suffices to show:
:$S_n x \to x$ for each $x \in X$.
Let $\sequence {x_n^\ast}_{n \mathop \in \N}$ be the [[Definition:Coordinate Functionals Associated with Schauder Basis|coordinate functionals associated with $\sequence {x_n}_{n \mathop \in \N... | Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology | https://proofwiki.org/wiki/Projections_Associated_with_Schauder_Basis_Converge_to_Identity_in_Strong_Operator_Topology | https://proofwiki.org/wiki/Projections_Associated_with_Schauder_Basis_Converge_to_Identity_in_Strong_Operator_Topology | [
"Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology",
"Projections Associated with Schauder Bases",
"Strong Operator Topology",
"Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology"
] | [
"Definition:Normed Vector Space",
"Definition:Schauder Basis",
"Definition:Projections Associated with Schauder Basis",
"Definition:Convergent Sequence",
"Definition:Identity Mapping",
"Definition:Strong Operator Topology"
] | [
"Characterization of Convergence in Strong Operator Topology",
"Definition:Coordinate Functionals Associated with Schauder Basis",
"Definition:Schauder Basis",
"Category:Projections Associated with Schauder Bases",
"Category:Strong Operator Topology",
"Category:Projections Associated with Schauder Basis C... |
proofwiki-22694 | Schwarz-Pick Theorem | Let $\Bbb D$ be the unit disk in the complex plane centered at $0$.
Let $f: \Bbb D \to \Bbb D$ be a holomorphic function.
Then:
:$\forall z \in \Bbb D: \dfrac {\cmod {\map {f'} z} } {1 - \cmod {\map f z}^2} \le \dfrac 1 {1 - \cmod z^2}$
where $\cmod z$ denotes the modulus of $z$.
Moreover, if there exists a $z_0 \in \B... | === Lemma ===
Let $z \in \Bbb D$ be arbitrary.
Let $\varphi_z$ be the Möbius transformation defined as:
:$\map {\varphi_z} w := \dfrac {w - z} {1 - \overline z w}$
where $\overline z$ denotes the complex conjugate of $z$.
Then $\varphi_z$ is a holomorphic bijection from $\Bbb D$ to $\Bbb D$ such that:
:its inverse is $... | Let $\Bbb D$ be the [[Definition:Unit Disk|unit disk]] in the [[Definition:Complex Plane|complex plane]] centered at $0$.
Let $f: \Bbb D \to \Bbb D$ be a [[Definition:Holomorphic Function|holomorphic function]].
Then:
:$\forall z \in \Bbb D: \dfrac {\cmod {\map {f'} z} } {1 - \cmod {\map f z}^2} \le \dfrac 1 {1 - \cm... | === Lemma ===
Let $z \in \Bbb D$ be [[Definition:arbitrary|arbitrary]].
Let $\varphi_z$ be the [[Definition:Möbius Transformation|Möbius transformation]] defined as:
:$\map {\varphi_z} w := \dfrac {w - z} {1 - \overline z w}$
where $\overline z$ denotes the [[Definition:Complex Conjugate|complex conjugate]] of $z$.
... | Schwarz-Pick Theorem | https://proofwiki.org/wiki/Schwarz-Pick_Theorem | https://proofwiki.org/wiki/Schwarz-Pick_Theorem | [] | [
"Definition:Unit Disk",
"Definition:Complex Number/Complex Plane",
"Definition:Holomorphic Function",
"Definition:Complex Modulus",
"Definition:Möbius Transformation"
] | [
"Definition:arbitrary",
"Definition:Möbius Transformation",
"Definition:Complex Conjugate",
"Definition:Holomorphic Function",
"Definition:Bijection",
"Definition:Inverse of Mapping",
"Definition:Derivative",
"Definition:Holomorphic Function",
"Definition:Holomorphic Function",
"Definition:Bijecti... |
proofwiki-22695 | Parity Check Detects Single Error | Let $\LL$ be a '''linear $\tuple {n, 2}$-code''' with a parity check.
Then $\LL$ is such that one transmission error will be detected. | Let $w$ be a transmitted codeword from $\LL$.
Let $s$ be the sum of the bits of $w$.
Let $w'$ be the received word corresponding to $w$.
Let $s'$ be the sum of the bits of $w'$.
Suppose that $w'$ has one transmission error.
Then either $w'$ has a $1$ instead of $0$, or $0$ instead of $1$.
In the first case, $s' = s + 1... | Let $\LL$ be a '''[[Definition:Linear Code|linear $\tuple {n, 2}$-code]]''' with a [[Definition:Parity Check|parity check]].
Then $\LL$ is such that one [[Definition:Transmission Error|transmission error]] will be detected. | Let $w$ be a [[Definition:Transmitted Codeword|transmitted codeword]] from $\LL$.
Let $s$ be the [[Definition:Integer Addition|sum]] of the [[Definition:Bit|bits]] of $w$.
Let $w'$ be the [[Definition:Received Word|received word]] corresponding to $w$.
Let $s'$ be the [[Definition:Integer Addition|sum]] of the [[Def... | Parity Check Detects Single Error | https://proofwiki.org/wiki/Parity_Check_Detects_Single_Error | https://proofwiki.org/wiki/Parity_Check_Detects_Single_Error | [
"Parity Checks"
] | [
"Definition:Linear Code",
"Definition:Parity Check",
"Definition:Transmission Error"
] | [
"Definition:Transmitted Codeword",
"Definition:Addition/Integers",
"Definition:Bit",
"Definition:Received Word",
"Definition:Addition/Integers",
"Definition:Bit",
"Definition:Transmission Error",
"Definition:Parity",
"Definition:Parity",
"Definition:Parity",
"Definition:Parity",
"Definition:Co... |
proofwiki-22696 | Space in which Every Closed Set is a Zero Set is T4 | Let $T = \struct{S, \tau}$ be a topological space such that every closed set is a zero set.
Then $T$ is a $T_4$ space. | Let $C, D \subseteq S$ be disjoint closed sets.
By hypothesis, there exist continuous functions:
:$f: S \to \closedint 0 1$ such that $C = f^{-1} (\set 0)$
:$g: S \to \closedint 0 1$ such that $D = g^{-1} (\set 0)$
Define $h: S \to \closedint 0 1$ by:
:$\map h x := \dfrac{\map f x}{\map f x + \map g x}$
Since $f(x), g(... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]] such that every [[Definition:Closed Set (Topology)|closed set]] is a [[Definition:Zero Set|zero set]].
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | Let $C, D \subseteq S$ be disjoint [[Definition:Closed Set (Topology)|closed sets]].
By hypothesis, there exist [[Definition:Continuous Mapping (Topology)|continuous functions]]:
:$f: S \to \closedint 0 1$ such that $C = f^{-1} (\set 0)$
:$g: S \to \closedint 0 1$ such that $D = g^{-1} (\set 0)$
Define $h: S \to \clo... | Space in which Every Closed Set is a Zero Set is T4 | https://proofwiki.org/wiki/Space_in_which_Every_Closed_Set_is_a_Zero_Set_is_T4 | https://proofwiki.org/wiki/Space_in_which_Every_Closed_Set_is_a_Zero_Set_is_T4 | [
"T4 Spaces",
"Zero Sets",
"Urysohn Functions"
] | [
"Definition:Topological Space",
"Definition:Closed Set/Topology",
"Definition:Zero Set",
"Definition:T4 Space"
] | [
"Definition:Closed Set/Topology",
"Definition:Continuous Mapping (Topology)",
"Definition:Urysohn Function",
"Urysohn's Lemma Converse"
] |
proofwiki-22697 | Seminorm on Vector Space induced by Linear Transformation is Seminorm | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be vector spaces over $\GF$.
Let $p$ be a seminorm on $Y$.
Define:
:$\map q x = \map p {T x}$
for each $x \in X$.
Then $q$ is a seminorm on $X$. | === Proof of $(\text N 2)$ ===
Let $x \in X$ and $\lambda \in \GF$.
We have:
{{begin-eqn}}
{{eqn | l = \map q {\lambda x}
| r = \map p {\map T {\lambda x} }
}}
{{eqn | r = \map p {\lambda T x}
| c = {{Defof|Linear Transformation}}
}}
{{eqn | r = \cmod \lambda \map p {T x}
| c = {{NormAxiomVector|2}} for $p$
}}
{{... | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be [[Definition:Vector Space|vector spaces]] over $\GF$.
Let $p$ be a [[Definition:Seminorm|seminorm]] on $Y$.
Define:
:$\map q x = \map p {T x}$
for each $x \in X$.
Then $q$ is a [[Definition:Seminorm|seminorm]] on $X$. | === Proof of $(\text N 2)$ ===
Let $x \in X$ and $\lambda \in \GF$.
We have:
{{begin-eqn}}
{{eqn | l = \map q {\lambda x}
| r = \map p {\map T {\lambda x} }
}}
{{eqn | r = \map p {\lambda T x}
| c = {{Defof|Linear Transformation}}
}}
{{eqn | r = \cmod \lambda \map p {T x}
| c = {{NormAxiomVector|2}} for $p$
}}
... | Seminorm on Vector Space induced by Linear Transformation is Seminorm | https://proofwiki.org/wiki/Seminorm_on_Vector_Space_induced_by_Linear_Transformation_is_Seminorm | https://proofwiki.org/wiki/Seminorm_on_Vector_Space_induced_by_Linear_Transformation_is_Seminorm | [
"Seminorms"
] | [
"Definition:Vector Space",
"Definition:Seminorm",
"Definition:Seminorm"
] | [] |
proofwiki-22698 | Finite Supremum of Family of Seminorms is Seminorm | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $\PP$ be a set of seminorms on $X$ such that:
:$\ds \sup_{p \in \PP} \map p x < \infty$ for each $x \in X$.
Define $q : X \to \R$ by:
:$\ds \map q x = \sup_{p \in \PP} \map p x$ for each $x \in X$.
Then $q$ is a seminorm on $X$. | === Proof of $(\text N 2)$ ===
Let $x \in X$ and $\lambda \in \GF$.
We have:
{{begin-eqn}}
{{eqn | l = \map q {\lambda x}
| r = \sup_{p \in \PP} \map p {\lambda x}
}}
{{eqn | o = \le
| r = \sup_{p \in \PP} \cmod \lambda \map p x
| c = {{NormAxiomVector|2}} for $p$
}}
{{eqn | r = \cmod \lambda \sup_{p \in \PP} \ma... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $\PP$ be a [[Definition:Set|set]] of [[Definition:Seminorm|seminorms]] on $X$ such that:
:$\ds \sup_{p \in \PP} \map p x < \infty$ for each $x \in X$.
Define $q : X \to \R$ by:
:$\ds \map q x = \sup_{p \in \PP} \map p ... | === Proof of $(\text N 2)$ ===
Let $x \in X$ and $\lambda \in \GF$.
We have:
{{begin-eqn}}
{{eqn | l = \map q {\lambda x}
| r = \sup_{p \in \PP} \map p {\lambda x}
}}
{{eqn | o = \le
| r = \sup_{p \in \PP} \cmod \lambda \map p x
| c = {{NormAxiomVector|2}} for $p$
}}
{{eqn | r = \cmod \lambda \sup_{p \in \PP} \... | Finite Supremum of Family of Seminorms is Seminorm | https://proofwiki.org/wiki/Finite_Supremum_of_Family_of_Seminorms_is_Seminorm | https://proofwiki.org/wiki/Finite_Supremum_of_Family_of_Seminorms_is_Seminorm | [
"Seminorms"
] | [
"Definition:Vector Space",
"Definition:Set",
"Definition:Seminorm",
"Definition:Seminorm"
] | [
"Multiple of Supremum"
] |
proofwiki-22699 | Complete Norms on Vector Space Equivalent iff Identity Mapping Bounded | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be norms on $X$ such that:
:$\struct {X, \norm {\, \cdot \,}_i}$ is a Banach space for $i \in \set {1, 2}$.
Let $I : \struct {X, \norm {\, \cdot \,}_1} \to \struct {X, \norm {\, \cdot \,}_2}$ be th... | === Necessary Condition ===
Suppose that:
:$\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are equivalent.
Then there exists $K > 0$ such that:
:$\norm {I x}_2 = \norm x_2 \le K \norm x_1$
That is:
:$I : \struct {X, \norm {\, \cdot \,}_1} \to \struct {X, \norm {\, \cdot \,}_2}$ is bounded.
{{qed|lemma}} | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be [[Definition:Norm on Vector Space|norms]] on $X$ such that:
:$\struct {X, \norm {\, \cdot \,}_i}$ is a [[Definition:Banach Space|Banach space]] for $i \in \set {1, ... | === Necessary Condition ===
Suppose that:
:$\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are [[Definition:Equivalence of Norms|equivalent]].
Then there exists $K > 0$ such that:
:$\norm {I x}_2 = \norm x_2 \le K \norm x_1$
That is:
:$I : \struct {X, \norm {\, \cdot \,}_1} \to \struct {X, \norm {\, \cdot \,}_2... | Complete Norms on Vector Space Equivalent iff Identity Mapping Bounded | https://proofwiki.org/wiki/Complete_Norms_on_Vector_Space_Equivalent_iff_Identity_Mapping_Bounded | https://proofwiki.org/wiki/Complete_Norms_on_Vector_Space_Equivalent_iff_Identity_Mapping_Bounded | [
"Banach Spaces"
] | [
"Definition:Vector Space",
"Definition:Norm/Vector Space",
"Definition:Banach Space",
"Definition:Identity Mapping",
"Definition:Equivalence of Norms",
"Definition:Bounded Linear Transformation"
] | [
"Definition:Equivalence of Norms",
"Definition:Bounded Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Bounded Linear Transformation"
] |
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