16.4: Convergencia uniforme de secuencias de funciones
- Page ID
- 86292
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Convergencia uniforme de secuencias de funciones
Para esta discusión, solo consideraremos funciones con\(g_n\) donde
\[\mathbb{R} \rightarrow \mathbb{R} \nonumber \]
Definición: Convergencia Uniforme
La secuencia (Sección 16.2)\(\left.\left\{g_{n}\right\}\right|_{n=1} ^{\infty}\) converge uniformemente para funcionar\(g\) si por cada\(\varepsilon > 0\) hay un entero\(N\) tal que\(n \geq N\) implica
\[\left|g_{n}(t)-g(t)\right| \leq \epsilon \label{16.11} \]
para todos\(t \in \mathbb{R}\).
Obviamente, cada secuencia uniformemente convergente es convergente puntual (Sección 16.3). La diferencia entre convergencia puntual y uniforme es la siguiente: Si\(\left\{g_{n}\right\}\) converge puntualmente a\(g\), entonces para cada\(\varepsilon> 0\) y para cada\(t \in \mathbb{R}\) hay un entero\(N\) dependiendo de\(\varepsilon\) y\(t\) tal que Ecuación\ ref {16.11} mantiene if\(n≥N\). Si\(\left\{g_{n}\right\}\) converge uniformemente a\(g\), es posible que cada\(\varepsilon>0\) uno encuentre un entero\(N\) que servirá para todos\(t \in \mathbb{R}\).
Ejemplo\(\PageIndex{1}\)
\[g_{n}(t)=\frac{1}{n}, t \in \mathbb{R} \nonumber \]
Dejemos\(\varepsilon > 0\) que se den. Entonces elige\(N=\left\lceil\frac{1}{\varepsilon}\right\rceil\). Obviamente,
\[\left|g_{n}(t)-0\right| \leq \epsilon, \quad n \geq N \nonumber \]
para todos\(t\). Así,\(g_n(t)\) converge de manera uniforme a\(0\).
Ejemplo\(\PageIndex{2}\)
\[g_{n}(t)=\frac{t}{n}, t \in \mathbb{R} \nonumber \]
Obviamente para cualquiera no\(\varepsilon > 0\) podemos encontrar una sola función\(g_n(t)\) para la que la Ecuación\ ref {16.11} se mantenga con\(g(t)=0\) para todos\(t\). Por lo tanto, no\(g_n\) es uniformemente convergente. Sin embargo sí tenemos:
\[g_{n}(t) \rightarrow g(t) \text{ pointwise }\nonumber \]
Conclusión
La convergencia uniforme siempre implica convergencia puntual, pero la convergencia puntual no garantiza una convergencia uniforme.