nyuuzyou/wikis huggingface dataset:
Smith Numbers are Infinite in Number/Lemma

== Theorem ==
Let $\map S m$ denote the sum of the digits of a positive integer $m$.
Let $\map {S_p} m$ denote the sum of the digits of the prime decomposition of $m$.
Let $\map N m$ denote the number of digits in $m$.
Suppose $m = p_1 p_2 \dots p_r$, where $p_i$ are prime numbers, not necessarily distinct.
Then $\map {S_p} m < 9 \map N m - 0.54 r$.
== Proof ==
Let $b_i = \map N {p_i} - 1, i = 1, 2, \dots, r$.
Let $b = b_1 + b_2 + \dots + b_r$.
Since a prime cannot be a multiple of $9$, $\map S {p_i}$ cannot be a multiple of $9$ either.
Hence:
:$\map S {p_i} \le 9 \map N {p_i} - 1 = 9 b_i + 8$
We partition the $p_i$ into $9$ disjoint classes by means of the following:
Define $c_i = \map S {p_i} - 9 b_i$.
By the above, $c_i \le 8$ and $c_i \ne 0$.
Let $n_0$ be the number of $i$ such that $c_i$ is (strictly) negative, and $n_j$ be the number of $i$ such that $c_i = j$ for $1 \le j \le 8$.
Then:
where the last sum is over the integers $i$ where $c_i$ is (strictly) negative.
There are $n_0$ of these integers, so we also have:
We have $\map S {p_i} = 9 b_i + c_i$.
For $c_i < 0$ we will use the inequality:
:$p_i > 10^{\map N {p_i} - 1} = 10^{b_i}$
For $1 \le c_i \le 8$ we would have:
:$p_i \ge \paren {c_i + 1} \times 10^{b_i} - 1$
since this number on the right is the smallest integer $N$ with $\map S N = 9 b_i + c_i$.
We also have:
:$\paren {c_i + 1} \times 10^{b_i} - 1 = \paren {c_i + 0.9} \times 10^{b_i} + 10^{b_i - 1} - 1 \ge \paren {c_i + 0.9} \times 10^{b_i}$
for $b_i > 0$ (i.e. when $p_i \ne 2, 3, 5, 7$).
Finally for $b_i = 0$, we have:
:$p_i = \map S {p_i} = c_i = c_i 10^{b_i}$
It follows that:
Write $m = a \times 10^{\map N m - 1}$ where $1 \le a < 10$.
By taking logarithm (base $10$) on both sides, we get:
Combining $(1)$ and $(2)$, we have:
== Sources ==
* 
Category:Smith Numbers are Infinite in Number
Smooth Local Coordinates for Product Manifold

== Theorem ==
Let $M_1, M_2$ be Riemannian manifolds.
Let $\tuple {x^1, \dots, x^n}$ be the smooth local coordinates for $M_1$.
Let $\tuple {x^{n + 1}, \dots, x^{n + m}}$ be the smooth local coordinates for $M_1$.
Let $M_1 \times M_2$ be the product manifold.
Then the smooth local coordinates for $M_1 \times M_2$ can be written as $\tuple {x^1, \dots, x^{n + m}}$.
== Proof ==
== Sources ==
* : $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics
Category:Riemannian Geometry
Smooth Local Parametrization of Surface of Revolution

== Theorem ==
Let $S_C$ and $C$ be the surface of revolution and its generating curve.
Let the smooth local parametrization of $C$ be:
:$\map \gamma t = \tuple {\map x t, \map y t}$
Then the smooth local parametrization of $S_C$ can be written as:
:$\map X {t, \theta} = \tuple {\map y t \cos \theta, \map y t \sin \theta, \map x t}$
where $\tuple {t, \theta}$ belongs to a sufficiently small open set in the plane, and the revolution is done around the $x$-axis.
== Proof ==
== Sources ==
* : $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics
Category:Surfaces of Revolution
Category:Solid Geometry
Smooth Manifold admits Lorentzian Metric iff admits Rank-1 Tangent Distribution

== Theorem ==
Let $M$ be a smooth manifold.
Then $M$ admits a Lorentzian metric $M$ admits a rank-$1$ tangent distribution.
That is, $M$ admits a Lorentzian metric $M$ admits a rank-$1$ subbundle of the tangent bundle $TM$.
== Proof ==
== Sources ==
* : $\S 2$: Riemannian Metrics. Pseudo-Riemannian Metrics
Category:Definitions/Pseudo-Riemannian Geometry
Smooth Mapping between Equidimensional Riemannian Manifolds is Local Isometry iff it is Isometry

== Theorem ==
Let $\struct {M, g}$ and $\struct {\tilde M, \tilde g}$ be Riemannian manifolds of the same dimension.
Let $\phi : M \to \tilde M$ be a smooth mapping.
Then $\phi$ is a local isometry it is an isometry, that is, $\phi^* \tilde g = g$.
== Proof ==
== Sources ==
* : $\S 2$: Riemannian Metrics. Definitions
Category:Isometries (Riemannian Manifolds)
Smooth Real Function times Derivative of Dirac Delta Distribution

== Theorem ==
Let $\alpha \in \map {C^\infty} \R$ be a smooth real function.
Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Then in the distributional sense it holds that:
:$\alpha \cdot \delta' = \map \alpha 0 \delta' - \map {\alpha'} 0 \delta$
== Proof ==
Let $\phi \in \map \DD \R$ be a test function.
== Sources ==
* : Chapter $\S 6.4$: A glimpse of distribution theory. Multiplication by $C^\infty$ functions
Category:Dirac Delta Function
Smooth Real Function times Derivative of Dirac Delta Distribution/Corollary

== Theorem ==
Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Then in the distributional sense it holds that:
:$x \delta' = - \delta$
== Proof ==
From Smooth Real Function times Derivative of Dirac Delta Distribution:
:$\alpha \cdot \delta' = \map \alpha 0 \delta' - \map {\alpha'} 0 \delta$
where $\alpha$ is a smooth function.
If $\map \alpha x = x$, then:
:$x \delta' = - \delta$
== Sources ==
* : Chapter $\S 6.4$: A glimpse of distribution theory. Multiplication by $C^\infty$ functions
Category:Dirac Delta Function
Smooth Vector Field as Sum of Smooth Horizontal and Vertical Vector Fields

== Theorem ==
Let $\tilde M, M$ be smooth manifolds.
Let $\pi : \tilde M \to M$ be a smooth submersion.
Let $\tilde g$ be a Riemannian metric on $\tilde M$.
Let $W$ be a smooth vector field on $\tilde M$.
Then $W$ can be uniquely decomposed as a sum:
:$W = W^H + W^V$
where $W^H$ and $W^V$ are smooth horizontal and vertical vector fields.
== Proof ==
== Sources ==
* : $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics
Category:Riemannian Geometry
Smooth Vector Field has Unique Smooth Horizontal Lift

== Theorem ==
Let $\tilde M, M$ be smooth manifolds.
Let $\pi : \tilde M \to M$ be a smooth submersion.
Let $\tilde g$ be a Riemannian metric on $\tilde M$.
Let $W$ be a smooth vector field on $M$.
Then $W$ has the unique smooth horizontal lift to $\tilde M$.
== Proof ==
== Sources ==
* : $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics
Category:Riemannian Geometry
Smullyan's Drinking Principle

== Theorem ==
Suppose that there is at least one person in the pub.
Then there is a person $x$ in the pub such that if $x$ is drinking, then everyone in the pub is drinking.
=== Formal Exposition ===
== Informal Proof ==
== Semi-Formal Proof ==
== Formal Proof ==
== Paradoxical Aspects ==
== Also see ==
* Paradoxes of Material Implication
Category:Veridical Paradoxes
Category:Logic
Category:Smullyan's Drinking Principle
Smullyan's Drinking Principle/Formal Exposition

== Formal Exposition of Smullyan's Drinking Principle ==
Let the universe of discourse be the non-empty set of people in the pub.
Let $\map D x$ be interpreted as the statement "$x$ is drinking".
Then:
:$\exists x : \paren {\map D x \implies \forall y : \map D y}$
== Proof ==
Category:Smullyan's Drinking Principle
Smullyan's Drinking Principle/Formal Proof

== Formal Exposition of Smullyan's Drinking Principle ==
== Proof ==
We have two choices:
:$\forall y : \map D y$
and
:$\neg \forall y : \map D y$
Suppose $\forall y : \map D y$.
By True Statement is implied by Every Statement:
:$\map D x \implies \forall y : \map D y$
By Existential Generalisation:
:$\exists x : \paren {\map D x \implies \forall y : \map D y}$
Now suppose:
:$\neg \forall y : \map D y$
By De Morgan's Laws (Predicate Logic)/Denial of Universality:
:$\exists y : \neg \map D y$
Switch the variable $y$ with $x$.
Thus, for some $x$:
:$\neg \map D x$
By False Statement implies Every Statement, we have:
:$\map D x \implies \forall y : \map D y$
By Existential Generalisation:
:$\exists x : \paren {\map D x \implies \forall y : \map D y}$
Thus, $\exists x : \paren {\map D x \implies \forall y : \map D y}$ holds both when:
:$\forall y : \map D y$
and when:
:$\neg \forall y : \map D y$
concluding the proof.
Category:Smullyan's Drinking Principle
Smullyan's Drinking Principle/Informal Proof

== Theorem ==
== Informal Proof ==
There are two cases:
:$(1): \quad$ Everyone in the pub is drinking.
:$(2): \quad$ Someone in the pub is not drinking.
Suppose first that everyone in the pub is drinking.
Then $x$ can be chosen to be any person in the pub.
Suppose instead that someone in the pub is not drinking.
Then $x$ can be chosen to be any person in the pub who is not drinking.
Category:Smullyan's Drinking Principle
Smullyan's Drinking Principle/Paradoxical Aspects

== Paradoxical Aspects to Smullyan's Drinking Principle ==
Smullyan's Drinking Principle is often considered to be a counter-intuitive result.
Hence it is often classified as a veridical paradox.
However, it can be resolved by analyzing the possible scenarios.
The semi-formal proof attempts to do just that.
Thus, a careful analysis of the possible scenarios, as well as certain Paradoxes of Material Implication, show this to be a veridical paradox.
Note that this is a theorem for arbitrary non-empty universes of discourse and arbitrary propositional functions $\map D x$.
So this theorem is valid for all interpretations of $\map D x$.
For example:
:let the universe of discourse be the objects in Philadelphia
:let $\map D x$ mean:
::$x$ is a dog
:Then there exists something in Philadelphia such that, if it is a dog, then everything in Philadelphia is a dog.
Category:Smullyan's Drinking Principle
Smullyan's Drinking Principle/Semi-Formal Proof

== Theorem ==
== Proof ==
Either everyone in the pub is drinking or someone in the pub is not drinking.
Suppose that everyone in the pub is drinking.
By True Statement is implied by Every Statement, the statement:
:everyone in the pub is drinking
is implied by the statement:
:$x$ is drinking
for any $x$ in the pub.
Since the pub is by assumption non-empty, there exists some $x$ in the pub.
Thus, there is some $x$ in the pub such that, if $x$ is drinking, then everyone in the pub is drinking.
Now suppose that there is some $x$ in the pub who is not drinking.
By False Statement implies Every Statement, if $x$ is drinking then everyone in the pub is drinking.
Thus, there is some $x$ in the pub such that, if $x$ is drinking, then everyone in the pub is drinking.
Category:Smullyan's Drinking Principle
Snake Lemma

== Theorem ==
Let $A$ be a commutative ring with unity.
Let:
::$\begin{xy}\xymatrix@L+2mu@+1em{
&
 M_1 \ar[r]_*{\alpha_1}
 \ar[d]^*{\phi_1}
&
 M_2 \ar[r]_*{\alpha_2}
 \ar[d]^*{\phi_2}
&
 M_3 \ar[d]^*{\phi_3}
 \ar[r]
& 
 0
\\
 0 \ar[r]
&
 N_1 \ar[r]_*{\beta_1}
&
 N_2 \ar[r]_*{\beta_2}
&
 N_3
&
}\end{xy}$
be a commutative diagram of $A$-modules.
Suppose that the rows are exact.
Then we have a commutative diagram:
::$\begin{xy}\xymatrix@L+2mu@+1em{
&
 \ker \phi_1 \ar[r]_*{\tilde\alpha_1}
 \ar[d]^*{\iota_1}
&
 \ker \phi_2 \ar[r]_*{\tilde\alpha_2}
 \ar[d]^*{\iota_2}
&
 \ker \phi_3
 \ar[d]^*{\iota_3}
&
\\
&
 M_1 \ar[r]_*{\alpha_1}
 \ar[d]^*{\phi_1}
&
 M_2 \ar[r]_*{\alpha_2}
 \ar[d]^*{\phi_2}
&
 M_3 \ar[d]^*{\phi_3}
 \ar[r]
& 
 0
\\
 0 \ar[r]
&
 N_1 \ar[r]_*{\beta_1}
 \ar[d]^*{\pi_1}
&
 N_2 \ar[r]_*{\beta_2}
 \ar[d]^*{\pi_2}
&
 N_3 \ar[d]^*{\pi_3}
\\
&
 \operatorname{coker} \phi_1 \ar[r]_*{\bar\beta_1}
&
 \operatorname{coker} \phi_2 \ar[r]_*{\bar\beta_2}
&
 \operatorname{coker} \phi_3
&
}\end{xy}$
where, for $i = 1, 2, 3$ respectively $i = 1, 2$:
* $\ker \phi_i$ is the kernel of $\phi_i$
* $\operatorname{coker} \phi_i$ is the cokernel of $\phi_i$
* $\iota_i$ is the inclusion mapping
* $\pi_i$ is the quotient epimorphism
* $\tilde \alpha_i = \alpha_i {\restriction \ker \phi_i}$, the restriction of $\alpha_i$ to $\ker \phi_i$
* $\bar \beta_i$ is defined by:
::$\forall n_i + \Img {\phi_i} \in \operatorname {coker} \phi_i : \map {\bar \beta_i} {n_i + \Img {\phi_i} } = \map {\beta_i} {n_i} + \Img {\phi_{i + 1} }$
Moreover there exists a morphism $\delta : \ker \phi_3 \to \operatorname{coker} \phi_1$ such that we have an exact sequence:
::$\begin{xy}\xymatrix@L+2mu@+1em{
 \ker \phi_1 \ar[r]_*{\tilde\alpha_1}
&
 \ker \phi_2 \ar[r]_*{\tilde\alpha_2}
&
 \ker\phi_3 \ar[r]_*{\delta}
& 
 \operatorname{coker}\phi_1 \ar[r]_*{\bar\beta_1}
&
 \operatorname{coker}\phi_2 \ar[r]_*{\bar\beta_2}
&
 \operatorname{coker}\phi_3
}\end{xy}$
== Proof ==
Category:Homological Algebra
Category:Named Theorems
Snell-Descartes Law

== Physical Law ==
Consider a ray of light crossing the threshold between two media.
Let its speed:
:in medium $1$ be $v_1$
:in medium $2$ be $v_2$.
Let it meet the threshold at:
:an angle $\alpha_1$ from the vertical in medium 1
:an angle $\alpha_2$ from the vertical in medium 2.
Then the Snell-Descartes law states that:
:$\dfrac {\sin \alpha_1} {v_1} = \dfrac {\sin \alpha_2} {v_2}$
== Proof ==
The Snell-Descartes law can be derived from Fermat's Principle of Least Time as follows:
Let the ray of light travel from $A$ to $P$ in the medium $1$.
Then let it travel from $P$ to $B$ in medium $2$.
:
The total time $T$ required for that journey is:
:$T = \dfrac {\sqrt {a^2 + x^2} } {v_1} + \dfrac {\sqrt {b^2 + \paren {c - x}^2} } {v_2}$
from the geometry of the above diagram.
From Fermat's Principle of Least Time, this time will be a minimum.
From Derivative at Maximum or Minimum, we need:
:$\dfrac {\d T} {\d x} = 0$.
So:
:$\dfrac x {v_1 \sqrt {a^2 + x^2} } = \dfrac {c - x} {v_2 \sqrt {b^2 + \paren {c - x}^2} }$
which leads directly to:
:$\dfrac {\sin \alpha_1} {v_1} = \dfrac {\sin \alpha_2} {v_2}$
by definition of sine.
== Also presented as ==
== Also known as ==
== Historical Note ==
== Sources ==
* : $1$: The Nature of Differential Equations: $\S 6$: The Brachistochrone. Fermat and the Bernoullis
* : Chapter $\text {A}.13$: Fermat ()
* : Chapter $\text {B}.22$: Bernoulli's Solution of the Brachistochrone Problem
* 
* 
Category:Snell-Descartes Law
Category:Optics
Snell-Descartes Law/Also known as

== Snell-Descartes Law: Also known as == 
The Snell-Descartes law is also known as just Snell's law.
In France this law is called la loi de Descartes or loi de Snell-Descartes, while elsewhere it tends to be known as Snell's law, or Snell's law of refraction.
Some sources refer to it as the sine law but on this is used as a different form of the name law of sines.
== Sources ==
* : $1$: The Nature of Differential Equations: $\S 6$: The Brachistochrone. Fermat and the Bernoullis
* : Chapter $\text {A}.13$: Fermat ()
* : Chapter $\text {B}.22$: Bernoulli's Solution of the Brachistochrone Problem
* 
* 
Category:Snell-Descartes Law
Snell-Descartes Law/Also presented as

== Snell-Descartes Law: Also known as == 
The Snell-Descartes law can also be seen expressed in the form:
:$\dfrac {\sin \alpha_1} {\sin \alpha_2} = \dfrac {v_1} {v_2}$
== Sources ==
* : Chapter $\text {A}.13$: Fermat ()
Category:Snell-Descartes Law
Snell-Descartes Law/Historical Note

== Historical Note on Snell-Descartes Law ==
 made his discovery in $1621$.
 was first to publish it, which he did in $1637$. 's own work did not appear until $1703$ when included it in his Dioptrica.
However, it appears that ' own statement was stated incorrectly as $\dfrac {\sin \alpha_1} {\sin \alpha_2} = \dfrac {v_2} {v_1}$, and proved inadequately.
It may well indeed be that himself learned it from , who now generally holds the credit for its discovery.
It needs to be noted that this law had previously been discovered by several other scientists, including in $984$ and in $1602$.
== Sources ==
* : Chapter $\text {A}.11$: Descartes ()
* : Chapter $\text {A}.13$: Fermat ()
Category:Historical Notes
Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods

== Theorem ==
Let $\struct{S, \tau}$ be a topological space.
For each $x \in S$, let:
:$\map \UU x$ denote the system of open neighborhoods of $x$ 
Then $\struct{S, \tau}$ is a sober space :
:for each completely prime filter $\FF$ in the complete lattice $\struct{\tau, \subseteq}$:
::$\exists ! x \in S : \FF = \map \UU x$
== Proof ==
Recall by definition of system of open neighborhoods:
:$\forall x \in S : \map \UU x = \set{U \in \tau : x \in U}$
=== Necessary Condition ===
Let $\struct{S, \tau}$ be a sober space.
=== Sufficient Condition ===
== Also see ==
* Topology forms Complete Lattice
== Sources ==
* : Chapter $1$: Spaces and Lattices of Open Sets, $\S 1$ Sober spaces, Proposition $1.3.1$
Category:Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods
Category:Sober Spaces
Category:Systems of Open Neighborhoods
Category:Completely Prime Filters
Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods/Necessary Condition

== Theorem ==
Let $\struct{S, \tau}$ be a $\struct{S, \tau}$ is a sober space.
For each $x \in S$, let:
:$\map \UU x$ denote the system of open neighborhoods of $x$ 
Then:
:for each completely prime filter $\FF$ in the complete lattice $\struct{\tau, \subseteq}$:
::$\exists ! x \in S : \FF = \map \UU x$
== Proof ==
Recall by definition of system of open neighborhoods:
:$\forall x \in S : \map \UU x = \set{U \in \tau : x \in U}$
Let $\FF$ be a completely prime filter of $\struct{\tau, \subseteq}$.
Let $W = \bigcup \set{U \in \tau : U \notin \FF}$.
By the contrapositive statement of the definition of completely prime filter:
:$W \notin \FF$
From Completely Prime Filter Induces Meet-Irreducible Open Set:
:$W$ is a meet-irreducible open set
By definition of sober space:
:$\exists ! x \in S : W = S \setminus \set x^-$
We have:
We also have
Hence:
:$\forall U \in \tau : U \notin \FF \iff U \notin \map \UU x$
From Biconditional Equivalent to Biconditional of Negations:
:$\forall U \in \tau : U \in \FF \iff U \in \map \UU x$
By definition of set equality:
:$\FF = \map \UU x$
From System of Open Neighborhoods are Equal Iff Singleton Closures are Equal:
:$x$ is unique
The result follows.
== Also see ==
* Topology forms Complete Lattice
Category:Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods
Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods/Sufficient Condition

== Theorem ==
Let $\struct{S, \tau}$ be a topological space.
For each $x \in S$, let:
:$\map \UU x$ denote the system of open neighborhoods of $x$ 
For each completely prime filter $\FF$ in the complete lattice $\struct{\tau, \subseteq}$, let: 
:$\exists ! x \in S : \FF = \map \UU x$
Then $\struct{S, \tau}$ is a sober space.
== Proof ==
Recall by definition of system of open neighborhoods:
:$\forall x \in S : \map \UU x = \set{U \in \tau : x \in U}$
For each completely prime filter $\FF$ in the complete lattice $\struct{\tau, \subseteq}$, let: 
:$\exists ! x \in S : \FF = \map \UU x$
Let $W \in \tau$ be a meet-irreducible open set.
Let $\FF = \set{U \in \tau : U \nsubseteq W}$.
From Meet-Irreducible Open Set Induces Completely Prime Filter:
:$\FF$ is a completely prime filter
We have :
:$\exists ! x \in S : \FF = \map \UU x$
We have:
We also have:
By definition of set equality:
:$W = S \setminus \set x^-$
From System of Open Neighborhoods are Equal Iff Singleton Closures are Equal:
:$x$ is unique
The result follows.
== Also see ==
* Topology forms Complete Lattice
Category:Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods
Sober Space is T0 Space

== Theorem ==
Let $\struct {S, \tau}$ be a sober space.
Then:
:$\struct {S, \tau}$ is a $T_0$ (Kolmogorov) space
== Proof ==
Let $x, y \in S : x \ne y$.
From Complement of Singleton Closure is Meet-Irreducible:
:$S \setminus \set x^-$ is a meet-irreducible open set
By definition of a sober space:
:$S \setminus \set x^-$ is uniquely defined by $x$
Hence:
:$S \setminus \set x^- \ne S \setminus \set y^-$
From Equal Relative Complements iff Equal Subsets:
:$\set x^- \ne \set y^-$
Since $x, y$ were arbitrary, it follows that:
:$\forall x, y \in S : x \ne y \implies \set x^- \ne \set y^-$
From Characterization of T0 Space by Distinct Closures of Singletons:
:$\struct {S, \tau}$ is a $T_0$ (Kolmogorov) space
== Also see ==
* T2 Space is Sober Space
== Sources ==
* : Chapter $1$: Spaces and Lattices of Open Sets, $\S 1$ Sober spaces, Notes $1.2 (1)$
Category:Sober Spaces
Category:T0 Spaces
Sobolev Norm is Norm

== Theorem ==
Let $k \in \Z_+$ and $1 \le p \le \infty$.
Let $U \subset \R^n$ be an open set.
Then the Sobolev norm is a norm on the Sobolev space $\map {W^{k, p} } U$.
== Proof ==
=== ===
Let $u, v \in \map {W^{k, p} } U$.
From P-Seminorm of Function Zero iff A.E. Zero: 
:$\norm u_{\map {W^{k, p} } U} = 0$
for $u \in \map {W^{k, p} } U$ $u = 0$ almost everywhere.
So is satisfied.
=== ===
Let $u \in \map {W^{k,p}} U$ and $\lambda \in \R$. 
Then: 
So is satisfied.
=== ===
Let $u, v \in \map {W^{k,p}} U$.
Let $1\le p < \infty$.
Then:
So is satisfied.
== Sources ==
* $\S5.2$ Sobolev Spaces, $3.$ Elementary properties, Theorem 2 (Sobolev spaces as function spaces).
Category:Sobolev Spaces
Sobolev Space is Banach Space

== Theorem ==
Let $k \in \Z_{\ge 0}$.
Let $p \in \R_{\ge 1}$ or $p = \infty$.
Let $U \subset \R^n$ be an open set.
Then the Sobolev space $\map {W^{k, p} } U$ equipped with the Sobolev norm is a Banach space.
== Proof ==
By Sobolev Norm is Norm, it remains to show that $\map {W^{k, p} } U$ is complete.
Assume $\sequence {u_m}$ is a Cauchy sequence in $\map {W^{k, p} } U$.
Then for each multiindex $\size \alpha \le k$, $\sequence {D^\alpha u_m}$ is a Cauchy sequence in $\map {L^p} U$.
From the Riesz-Fischer Theorem, $\map {L^p} U$ is complete.
Hence $\sequence {D^\alpha u_m}$ converges in $L^p$ norm:
:$D^\alpha u_m \to u_\alpha\in \map {L^p} U$
for each $\size \alpha \le k$.
In particular, $\sequence{u_m}$ converges in $L^p$ norm:
:$u_m \to u_{\tuple {0, \ldots, 0} } =: u$
We now claim:
To verify this assertion, fix $\phi \in \map {C_c^\infty} U$.
Then:
By the definition of distributional derivative, $(1)$ holds.
Therefore:
:$D^\alpha u_m \to D^\alpha u$ in $\map {L^p} U$
for all $\size \alpha \le k$.
We see that:
:$u_m \to u$ in $\map {W^{k, p} } U$
as required.
== Sources ==
* $\S5.2$ Sobolev Spaces, $3.$ Elementary properties, Theorem 2 (Sobolev spaces as function spaces).
Category:Sobolev Spaces
Category:Banach Spaces
Sociable Chain/Examples/12,496

== Example of Sociable Chain ==
The sociable chain whose smallest element is $12 \, 496$ is of order $5$.
It goes:
:$12 \, 496 \to 14 \, 288 \to 15 \, 472 \to 14 \, 536 \to 14 \, 264 \to 12 \, 496$
== Proof ==
Let $\map s n$ denote the aliquot sum of $n$.
By definition:
:$\map s n = \map {\sigma_1} n - n$
where $\sigma_1$ denotes the divisor sum function.
Thus:
It is interesting to list the prime decomposition of each of the terms in the chain:
== Also see ==
* Sociable Chain of $14 \, 316$
== Historical Note ==
== Sources ==
* 
* : $28$
* : $12,496$
* : $28$
* : $12,496$
* 
Category:Sociable Numbers
Category:Sociable Chain of 12,496
Sociable Chain/Examples/12,496/Mistake

== Source Work ==
:
:The Dictionary
::$28$
:
:The Dictionary
::$28$
== Mistake ==
:The longest known sociable chain is of $28$ links, starting with $12,496$.
This number $12,496$ actually starts a chain of $5$ links.
The $28$-link chain actually starts at $14,316$.
== Sources ==
* : $28$
* : $28$
Category:Mistakes/Curious and Interesting Numbers
Sociable Chain/Examples/14,316

== Example of Sociable Chain ==
The longest known sociable chain, at time of writing ($9$th March $2017$), is of order $28$.
Its smallest element is $14 \, 316$.
== Proof ==
Let $\map s n$ denote the aliquot sum of $n$.
By definition:
:$\map s n = \map {\sigma_1} n - n$
where $\map {\sigma_1} n$ denotes the divisor sum function.
Thus:
It is interesting to list the prime decomposition of each of the terms in the chain:
== Also see ==
* Sociable Chain of $12 \, 496$
== Historical Note ==
== Sources ==
* 
* : $14,316$
* : $14,316$
* 
Category:Sociable Numbers
Category:Sociable Chain of 14,316
Sociable Chain/Examples/14,316/Historical Note

== Historical Note on Sociable Chain of 14,316 ==
The sociable chain of $14 \, 316$ was discovered by .
He included a reference to it in a paper he published in $1918$.
== Sources ==
* 
* : $12,496$
* : $14,316$
* : $12,496$
* : $14,316$
* 
Category:Historical Notes