Tue-27-05-2025 23:04
profe: Natalia Accomazzo Scotti
status:
tags: Espacios Normados
Def. : {\color{Cyan} \text{Def. :} } Def. :
Sea E espacio vectorial (ev), S ⊆ E subespacio. S es un hiperplano si ∃ x ∉ S tal que E = S + ⟨ x ⟩ \begin{array}{l}
\text{Sea $E$ espacio vectorial (ev), $S\subseteq E$ subespacio. $S$ es un hiperplano}\\
\text{si $\:\exists\:x\not\in S$ tal que $E=S+\braket{ x}$ }
\end{array} Sea E espacio vectorial (ev), S ⊆ E subespacio. S es un hiperplano si ∃ x ∈ S tal que E = S + ⟨ x ⟩ Prop. : {\color{Orange} \text{Prop. :} } Prop. :
S ⊆ E es hiperplano ⟺ ∃ φ : E → R lineal tal que S = ker ( φ ) \begin{array}{l}
\text{$S\subseteq E$ es hiperplano $\iff \:\exists\:{\Large\varphi}:E\to \mathbb{R}$ lineal tal que $S=\ker({\Large\varphi})$ }
\end{array} S ⊆ E es hiperplano ⟺ ∃ φ : E → R lineal tal que S = ker ( φ ) Prop. : {\color{Orange} \text{Prop. :} } Prop. :
Sea ( E , ∣ ∣ ⋅ ∣ ∣ ) ∣ dim ( E ) = n ⟹ E es isom e ˊ trico a R n \begin{array}{l}
\text{Sea $(E,\lvert \lvert \cdot \rvert\rvert)\bigm|\dim(E)=n\implies E$ es isométrico a $\mathbb{R}^{n}$ }
\end{array} Sea ( E , ∣∣ ⋅ ∣∣) dim ( E ) = n ⟹ E es isom e ˊ trico a R n Dem: {\color{Orange} \text{Dem:} } Dem:
T : E → R n v i ↦ e i \begin{array}{c}
T:E&\to& \mathbb{R}^{n} \\
v_{i}&\mapsto&e_{i}
\end{array} T : E v i → ↦ R n e i y extiendo por linealidad donde { v 1 , … , v n } \{ v_{1},\dots,v_{n} \} { v 1 , … , v n } E E E { e 1 , … , e n } \{ e_{1},\dots,e_{n} \} { e 1 , … , e n } R n \mathbb{R}^{n} R n R n \mathbb{R}^{n} R n
∣ ∣ ( x 1 , … x n ) ∣ ∣ R n = ∣ ∣ ∑ i = 1 n x i ⋅ v i ∣ ∣ E ⟹ T es isometr ı ˊ a con esta norma \lvert \lvert (x_{1},\dots x_{n}) \rvert\rvert _{\mathbb{R}^{n} }=\left\lvert \left\lvert \sum_{i=1}^{n} x_{i}\cdot v_{i} \right\rvert \right\rvert _{E}\implies T \text{ es isometría con esta norma} ∣∣( x 1 , … x n )∣ ∣ R n = i = 1 ∑ n x i ⋅ v i E ⟹ T es isometr ı ˊ a con esta norma T ( ∑ i = 1 n x i ⋅ v i ) = ∑ i = 1 n x i ⋅ T ( v i ) = ∑ i = 1 n x i ⋅ e i = ( x 1 , … , x n ) T\left( \sum_{i=1}^{n} x_{i}\cdot v_{i} \right)=\sum_{i=1}^{n} x_{i}\cdot T(v_{i})=\sum_{i=1}^{n} x_{i}\cdot e_{i}=(x_{1},\dots,x_{n}) T ( i = 1 ∑ n x i ⋅ v i ) = i = 1 ∑ n x i ⋅ T ( v i ) = i = 1 ∑ n x i ⋅ e i = ( x 1 , … , x n ) Q.E.D. □ \quad \quad \quad \quad \quad \quad \quad \quad \boxed{\text{Q.E.D.}} \quad \square Q.E.D. □ Corolario (1) : {\color{Red} \text{Corolario (1)}:} Corolario (1) :
Si dim ( E ) < ∞ ⟹ Todas las normas son equivalentes. \begin{array}{l}
\text{Si $\dim(E)<\infty\implies$ Todas las normas son equivalentes.}
\end{array} Si dim ( E ) < ∞ ⟹ Todas las normas son equivalentes. Dem : {\color{Red} \text{Dem}:} Dem :
Corolario (2) : {\color{Red} \text{Corolario (2)}:} Corolario (2) :
Si dim ( E ) < ∞ ⟹ es de Banach. \begin{array}{l}
\text{Si $\dim(E)<\infty\implies$ es de Banach.}
\end{array} Si dim ( E ) < ∞ ⟹ es de Banach. Dem : {\color{Red} \text{Dem}:} Dem : ( x n ) n ∈ N ( x_{n} )_{n \in \mathbb{N}} ( x n ) n ∈ N E ⟹ ( T ( x n ) ) n ∈ N E\implies(T(x_{n}))_{n\in \mathbb{N}} E ⟹ ( T ( x n ) ) n ∈ N R n ⟹ \mathbb{R}^{n}\implies R n ⟹ R n \mathbb{R}^{n} R n ⟹ ( x n ) n ∈ N \implies ( x_{n} )_{n \in \mathbb{N}} ⟹ ( x n ) n ∈ N
Corolario ( 3 ) : {\color{Red}\text{Corolario }(3):} Corolario ( 3 ) :
Si dim ( E ) < 0 ? ? ⟹ conjuntos cerrados y acotados son compactos. \begin{array}{l}
\text{Si $\dim(E)<0??\implies$ conjuntos cerrados y acotados son compactos.}
\end{array} Si dim ( E ) < 0 ?? ⟹ conjuntos cerrados y acotados son compactos. Dem : {\color{Red} \text{Dem}:} Dem :
Operadores lineales continuos
R e c o r d a r : Recordar: R ecor d a r : T : E → F T:E\to F T : E → F ∀ x , y ∈ E , α ∈ R : \quad\forall x,y \in E,\alpha \in \mathbb{R}: ∀ x , y ∈ E , α ∈ R :
T ( x + y ) = T ( x ) + T ( y ) T(x+y)=T(x)+T(y) T ( x + y ) = T ( x ) + T ( y ) T ( α ⋅ x ) = α ⋅ T ( x ) T(\alpha\cdot x)=\alpha\cdot T(x) T ( α ⋅ x ) = α ⋅ T ( x ) T ( 0 E ) = 0 F T(0_{E})=0_{F} T ( 0 E ) = 0 F
O b s : Obs: O b s :
T : R n → R n transformaci o ˊ n lineal ⟹ es continua. \text{$T:\mathbb{R}^{n}\to \mathbb{R}^{n}$ transformación lineal $\implies$ es continua.} T : R n → R n transformaci o ˊ n lineal ⟹ es continua. Esto se debe a que T T T A A A T ( x ) = A ⋅ x T(x)=A\cdot x T ( x ) = A ⋅ x ⟹ ∣ ∣ A ⋅ x − A ⋅ y ∣ ∣ = ∣ ∣ A ( x − y ) ∣ ∣ ≤ ∣ ∣ A ∣ ∣ ⋅ ∣ ∣ x − y ∣ ∣ ⟹ T \implies \lvert \lvert A\cdot x-A\cdot y \rvert\rvert=\lvert \lvert A(x-y) \rvert\rvert\leq \lvert \lvert A \rvert\rvert\cdot \lvert \lvert x-y \rvert\rvert\implies T ⟹ ∣∣ A ⋅ x − A ⋅ y ∣∣ = ∣∣ A ( x − y )∣∣ ≤ ∣∣ A ∣∣ ⋅ ∣∣ x − y ∣∣ ⟹ T
Pregunta 1: T : E → R n T:E\to \mathbb{R}^{n} T : E → R n ⟹ \implies ⟹
Pregunta 2: T : R n → E T:\mathbb{R}^{n}\to E T : R n → E ⟹ \implies ⟹
E j e m p l o Ejemplo E j e m pl o E = { p : R → R ∣ p polinomio de coef. reales } E=\{ p:\mathbb{R}\to \mathbb{R}\bigm| p\text{ polinomio de coef. reales} \} E = { p : R → R p polinomio de coef. reales }
( p + q ) ( x ) = p ( x ) + q ( x ) (p+q)(x)=p(x)+q(x) ( p + q ) ( x ) = p ( x ) + q ( x ) ( α ⋅ p ) ( x ) = α ⋅ p ( x ) (\alpha\cdot p)(x)=\alpha\cdot p(x) ( α ⋅ p ) ( x ) = α ⋅ p ( x )
∣ ∣ p ∣ ∣ = s u p x ∈ [ 0 , 1 ] ∣ p ( x ) ∣ \lvert \lvert p \rvert\rvert=\underset{ x \in[0,1] }{ sup }\:\left| p(x) \right| ∣∣ p ∣∣ = x ∈ [ 0 , 1 ] s u p ∣ p ( x ) ∣ ∣ ∣ p ∣ ∣ = 0 ⟹ ∣ p ( x ) ∣ = 0 ∀ x ∈ [ 0 , 1 ] \lvert \lvert p \rvert\rvert=0\implies |p(x)|=0\quad\forall x \in[0,1] ∣∣ p ∣∣ = 0 ⟹ ∣ p ( x ) ∣ = 0 ∀ x ∈ [ 0 , 1 ] p 0 = 0 p_{0}=0 p 0 = 0
T : E → R p ↦ p ( 2 ) \begin{array}{c}
T:E&\to &\mathbb{R} \\
p&\mapsto&p(2)
\end{array} T : E p → ↦ R p ( 2 ) T T T T T T p 0 = 0 p_{0}=0 p 0 = 0
Sea p n ( x ) = ( x 2 ) n . p_{n}(x)=\left( \frac{x}{2} \right)^{n}. p n ( x ) = ( 2 x ) n . p n ⟶ p 0 p_{n}\longrightarrow p_{0} p n ⟶ p 0
x ∈ [ 0 , 1 ] ⟹ ∣ p n ( x ) − p 0 ( x ) ∣ = ∣ x 2 ∣ n ≤ 1 2 n ⟶ 0 x \in [0,1]\implies |p_{n}(x)-p_{0}(x)|=\left| \frac{x}{2} \right| ^{n} \leq \frac{1}{2^{n} }\longrightarrow 0 x ∈ [ 0 , 1 ] ⟹ ∣ p n ( x ) − p 0 ( x ) ∣ = 2 x n ≤ 2 n 1 ⟶ 0 T ( p n ) = p n ( 2 ) = 1 ⟶ 0 T(p_{n})=p_{n}(2)=1\cancel{ \longrightarrow } 0 T ( p n ) = p n ( 2 ) = 1 ⟶ 0 Teorema : {\color{violet} \text{Teorema :} } Teorema :
T : E → F lineal. Son equivalentes: 1. T es continua en 0 2. T es continua en alg u ˊ n x ∈ E 3. T es continua (en todos lados) 4. T es uniformemente continua \begin{array}{l}
\text{$T:E\to F$ lineal. Son equivalentes:}\\
\text{1. $T$ es continua en $0$}\\
\text{2. $T$ es continua en algún $x \in E$}\\
\text{3. $T$ es continua (en todos lados)}\\
\text{4. $T$ es uniformemente continua}\\
\end{array} T : E → F lineal. Son equivalentes: 1. T es continua en 0 2. T es continua en alg u ˊ n x ∈ E 3. T es continua (en todos lados) 4. T es uniformemente continua Dem: {\color{violet} \text{Dem:} } Dem: ( 1 ) ⟹ ( 4 ) : (1)\implies(4): ( 1 ) ⟹ ( 4 ) : E > 0. \mathcal{E}>0. E > 0. ∃ δ > 0 ∣ ∣ ∣ x − y ∣ ∣ E < δ ⟹ ∣ ∣ T ( x ) − T ( y ) ∣ ∣ F < E \:\exists\:\delta>0\bigm|\lvert \lvert x-y \rvert\rvert_{E}<\delta\implies \lvert \lvert T(x)-T(y) \rvert\rvert_{F}<\mathcal{E} ∃ δ > 0 ∣∣ x − y ∣ ∣ E < δ ⟹ ∣∣ T ( x ) − T ( y )∣ ∣ F < E δ > 0 \delta>0 δ > 0 0. 0. 0.
∣ ∣ v ∣ ∣ E < δ 0 > ∣ ∣ T ( v ) − T ( 0 ) ∣ ∣ F < E \lvert \lvert v \rvert\rvert _{E}<\delta_{0}>\lvert \lvert T(v)-T(0) \rvert\rvert _{F}<\mathcal{E} ∣∣ v ∣ ∣ E < δ 0 > ∣∣ T ( v ) − T ( 0 )∣ ∣ F < E Si x , y ∈ E ∣ ∣ ∣ x − y ∣ ∣ E < δ ⟹ ∣ ∣ T ( x − y ) − T ( 0 ) ∣ ∣ F = ∣ ∣ T ( x ) − T ( y ) ∣ ∣ F < E x,y \in E\bigm|\lvert \lvert x-y \rvert\rvert_{E}<\delta\implies \lvert \lvert T(x-y)-T(0) \rvert\rvert_{F}=\lvert \lvert T(x)-T(y) \rvert\rvert_{F}<\mathcal{E} x , y ∈ E ∣∣ x − y ∣ ∣ E < δ ⟹ ∣∣ T ( x − y ) − T ( 0 )∣ ∣ F = ∣∣ T ( x ) − T ( y )∣ ∣ F < E ( 4 ) ⟹ ( 3 ) ⟹ ( 2 ) : (4)\implies(3)\implies(2): ( 4 ) ⟹ ( 3 ) ⟹ ( 2 ) :
( 2 ) ⟹ ( 1 ) : (2)\implies(1): ( 2 ) ⟹ ( 1 ) :
Sea x ∈ E ∣ T x \in E\bigm|T x ∈ E T x ⟹ ∀ E > 0 ∃ δ > 0 ∣ ∣ ∣ x − y ∣ ∣ E < δ ⟹ ∣ ∣ T ( x ) − T ( y ) ∣ ∣ F < E x\implies \forall\mathcal{E}>0\:\exists\:\delta>0\bigm|\lvert \lvert x-y \rvert\rvert_{E}<\delta\implies \lvert \lvert T(x)-T(y) \rvert\rvert_{F}<\mathcal{E} x ⟹ ∀ E > 0 ∃ δ > 0 ∣∣ x − y ∣ ∣ E < δ ⟹ ∣∣ T ( x ) − T ( y )∣ ∣ F < E ⟹ \implies ⟹ E > 0 \mathcal{E}>0 E > 0 δ > 0 \delta>0 δ > 0 x x x v ∈ B ( 0 , δ ) v \in B(0,\delta) v ∈ B ( 0 , δ ) ⟹ ∣ ∣ v ∣ ∣ E < δ \implies \lvert \lvert v \rvert\rvert_{E}<\delta ⟹ ∣∣ v ∣ ∣ E < δ y = v + x ⟹ ∣ ∣ x − y ∣ ∣ E < δ ⟹ y=v+x\implies \lvert \lvert x-y \rvert\rvert_{E}<\delta\implies y = v + x ⟹ ∣∣ x − y ∣ ∣ E < δ ⟹
⟹ ∣ ∣ T ( x ) − T ( y ) ∣ ∣ F < E = ∣ ∣ T ( x ) − T ( v ) − T ( x ) ∣ ∣ F < E = ∣ ∣ T ( v ) − T ( 0 ) ∣ ∣ F < E \begin{array}{l}
\implies\lvert \lvert T(x)-T(y) \rvert\rvert_{F} <\mathcal{E} \\
=\lvert \lvert T(x)-T(v)-T(x) \rvert\rvert _{F}<\mathcal{E} \\
=\lvert \lvert T(v)-T(0) \rvert\rvert _{F}<\mathcal{E}
\end{array} ⟹ ∣∣ T ( x ) − T ( y )∣ ∣ F < E = ∣∣ T ( x ) − T ( v ) − T ( x )∣ ∣ F < E = ∣∣ T ( v ) − T ( 0 )∣ ∣ F < E Q.E.D. □ \quad \quad \quad \quad \quad \quad \quad \quad \boxed{\text{Q.E.D.}} \quad \square Q.E.D. □ Def. : {\color{Cyan} \text{Def. :} } Def. :
Sea T : E → F una transformaci o ˊ n lineal. T es acotada si ∃ C > 0 ∣ ∣ ∣ T ( x ) ∣ ∣ F ≤ C ⋅ ∣ ∣ x ∣ ∣ E ∀ x ∈ E \begin{array}{l}
\text{Sea $T:E\to F$ una transformación lineal. }\\
\text{$T$ es acotada si $\:\exists\:C>0\bigm|\lvert \lvert T(x) \rvert\rvert_{F}\leq C\cdot \lvert \lvert x \rvert\rvert_{E}\quad\forall x \in E$ }
\end{array} Sea T : E → F una transformaci o ˊ n lineal. T es acotada si ∃ C > 0 ∣∣ T ( x )∣ ∣ F ≤ C ⋅ ∣∣ x ∣ ∣ E ∀ x ∈ E Equivalentemente, T T T s u p x ∈ B ( 0 , 1 ) ∣ ∣ T ( x ) ∣ ∣ F < ∞ \underset{ x \in B(0,1) }{ sup }\:\lvert \lvert T(x) \rvert\rvert_{F}<\infty x ∈ B ( 0 , 1 ) s u p ∣∣ T ( x )∣ ∣ F < ∞ D e m : Dem: D e m : ⟹ ) \implies) ⟹ )
x ∈ B ( 0 , 1 ) ⟹ ∣ ∣ T ( x ) ∣ ∣ F ≤ C ⋅ ∣ ∣ x ∣ ∣ E ≤ C ⟹ s u p x ∈ B ( 0 , 1 ) ∣ ∣ T ( x ) ∣ ∣ F ≤ C < ∞ x \in B(0,1)\implies \lvert \lvert T(x) \rvert\rvert _{F}\leq C\cdot \lvert \lvert x \rvert\rvert_{E}\leq C\implies \underset{ x \in B(0,1) }{ sup }\:\lvert \lvert T(x) \rvert\rvert _{F}\leq C<\infty x ∈ B ( 0 , 1 ) ⟹ ∣∣ T ( x )∣ ∣ F ≤ C ⋅ ∣∣ x ∣ ∣ E ≤ C ⟹ x ∈ B ( 0 , 1 ) s u p ∣∣ T ( x )∣ ∣ F ≤ C < ∞ ⟸ ) \impliedby) ⟸ ) x ∈ E . x \in E. x ∈ E . x = 0 x=0 x = 0 x ≠ 0 , x ~ = x ( 1 + E ) ⋅ ∣ ∣ x ∣ ∣ E x\neq0, \tilde{x}=\frac{x}{(1+\mathcal{E})\cdot \lvert \lvert x \rvert\rvert_{E}} x = 0 , x ~ = ( 1 + E ) ⋅ ∣∣ x ∣ ∣ E x E > 0 \mathcal{E}>0 E > 0
⟹ ∣ ∣ x ~ ∣ ∣ E = 1 1 + E < 1 ⟹ x ~ ∈ B ( 0 , 1 ) ⟹ ∣ ∣ T ( x ~ ) ∣ ∣ F ≤ C = s u p x ∈ B ( 0 , 1 ) ∣ ∣ T ( x ) ∣ ∣ F \implies \lvert \lvert \tilde{x} \rvert\rvert _{E}=\frac{1}{1+\mathcal{E}}<1\implies \tilde{x}\in B(0,1)\implies \lvert \lvert T(\tilde{x}) \rvert\rvert _{F}\leq C=\underset{ x \in B(0,1) }{ sup }\:\lvert \lvert T(x) \rvert\rvert _{F} ⟹ ∣∣ x ~ ∣ ∣ E = 1 + E 1 < 1 ⟹ x ~ ∈ B ( 0 , 1 ) ⟹ ∣∣ T ( x ~ )∣ ∣ F ≤ C = x ∈ B ( 0 , 1 ) s u p ∣∣ T ( x )∣ ∣ F ⟹ ∣ ∣ T ( x ( 1 + E ) ⋅ ∣ ∣ x ∣ ∣ E ) ∣ ∣ F = ∣ ∣ T ( x ) ∣ ∣ F ( 1 + E ) ⋅ ∣ ∣ x ∣ ∣ E ≤ C \implies \left\lvert \left\lvert T\left( \frac{x}{(1+\mathcal{E})\cdot \lvert \lvert x \rvert\rvert _{E}} \right) \right\rvert \right\rvert_{F}=\frac{\lvert \lvert T(x) \rvert\rvert _{F}}{(1+\mathcal{E})\cdot \lvert \lvert x \rvert\rvert _{E}}\leq C ⟹ T ( ( 1 + E ) ⋅ ∣∣ x ∣ ∣ E x ) F = ( 1 + E ) ⋅ ∣∣ x ∣ ∣ E ∣∣ T ( x )∣ ∣ F ≤ C ⟹ ∣ ∣ T ( x ) ∣ ∣ F ≤ C ⋅ ( 1 + E ) ⋅ ∣ ∣ x ∣ ∣ E ⟶ C ⋅ ∣ ∣ x ∣ ∣ E \implies \lvert \lvert T(x) \rvert\rvert _{F}\leq C\cdot(1+\mathcal{E})\cdot \lvert \lvert x \rvert\rvert _{E}\longrightarrow C\cdot \lvert \lvert x \rvert\rvert _{E} ⟹ ∣∣ T ( x )∣ ∣ F ≤ C ⋅ ( 1 + E ) ⋅ ∣∣ x ∣ ∣ E ⟶ C ⋅ ∣∣ x ∣ ∣ E Teorema : {\color{violet} \text{Teorema :} } Teorema :
Sea T : E → F transformaci o ˊ n lineal. T es acotada ⟺ T es continua. \begin{array}{l}
\text{Sea $T:E\to F$ transformación lineal. $T$ es acotada $\iff T$ es continua.}
\end{array} Sea T : E → F transformaci o ˊ n lineal. T es acotada ⟺ T es continua. Dem: {\color{violet} \text{Dem:} } Dem: ⟹ ) \implies) ⟹ ) T T T ⟹ T \implies T ⟹ T ⟹ T \implies T ⟹ T ⟹ T \implies T ⟹ T
⟸ ) \impliedby) ⟸ ) T T T 0. 0. 0. E = 1 \mathcal{E}=1 E = 1 ∃ δ > 0 ∣ x ∈ B ( 0 , δ ) ⟹ ∣ ∣ T ( x ) ∣ ∣ F < 1 \:\exists\:\delta>0\bigm|x\in B(0,\delta)\implies \lvert \lvert T(x) \rvert\rvert_{F}<1 ∃ δ > 0 x ∈ B ( 0 , δ ) ⟹ ∣∣ T ( x )∣ ∣ F < 1 y ∈ B ( 0 , 1 ) ⟹ δ ⋅ y ∈ B ( 0 , δ ) ⟹ ∣ ∣ T ( δ ⋅ y ) ∣ ∣ F < 1 y \in B(0,1)\implies\delta\cdot y \in B(0,\delta)\implies \lvert \lvert T(\delta\cdot y) \rvert\rvert_{F}<1 y ∈ B ( 0 , 1 ) ⟹ δ ⋅ y ∈ B ( 0 , δ ) ⟹ ∣∣ T ( δ ⋅ y )∣ ∣ F < 1 ⟹ ∣ ∣ T ( y ) ∣ ∣ F < 1 δ \implies \lvert \lvert T(y) \rvert\rvert_{F}<\frac{1}{\delta} ⟹ ∣∣ T ( y )∣ ∣ F < δ 1
Q.E.D. □ \quad \quad \quad \quad \quad \quad \quad \quad \boxed{\text{Q.E.D.}} \quad \square Q.E.D. □ Prop. : {\color{Orange} \text{Prop. :} } Prop. :
E , F normados, F de Banach, D ⊆ E subespacio denso. T : D → F t.f. continua ⟹ ∃ T ~ : E → T lineal y continua con T ~ ∣ D = T \begin{array}{l}
\text{$E,F$ normados, $F$ de Banach, $D\subseteq E$ subespacio denso.}\\
\text{$T:D\to F$ t.f. continua $\implies \:\exists\:\tilde{T}:E\to T$ lineal y continua con $\tilde{T}|_{_{D}}=T$}
\end{array} E , F normados, F de Banach, D ⊆ E subespacio denso. T : D → F t.f. continua ⟹ ∃ T ~ : E → T lineal y continua con T ~ ∣ D = T Dem: {\color{Orange} \text{Dem:} } Dem: x ∈ E ⟹ ∃ ( x n ) n ∈ N ⊆ D x \in E\implies \:\exists\:( x_{n} )_{n \in \mathbb{N}}\subseteq D x ∈ E ⟹ ∃ ( x n ) n ∈ N ⊆ D x n ⟶ x x_{n}\longrightarrow x x n ⟶ x T ~ ( x ) = lim n → ∞ T ( x n ) \tilde{T}(x)=\underset{ n\to \infty }{ \lim }T(x_{n}) T ~ ( x ) = n → ∞ lim T ( x n )
Citas y Comentarios 
de Cauchy en de Cauchy en converge en converge.