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\usepackage{amsmath,amsthm,amssymb,amsfonts, enumitem, fancyhdr, color, comment, graphicx, environ}

\usepackage[brazilian]{babel}
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\newenvironment{problem}[2][Exerc\'icio]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{sol}
    {\emph{Solution:}
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    {
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\newcommand{\ben}{\begin{enumerate}[label=\alph*.]}
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%Fill in the appropriate information below
\lhead{Lista de Exercícios 5}  %replace with your name
\rhead{Análise Funcional \\ 1º Quadrimestre de 2024 \\ Teoremas fundamentais I} %replace XYZ with the homework course number, semester (e.g. ``Spring 2019"), and assignment number.
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\begin{document}
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\pagenumbering{arabic}

%Solutions to problems go below.  Please follow the guidelines from https://www.overleaf.com/read/sfbcjxcgsnsk/


%Copy the following block of text for each problem in the assignment.


\begin{problem}{1}
Mostre que para todo espaço normado de dimensão infinita $U$ e todo espaço normado $V\neq \{0\}$, existe um operador linear descontínuo $T:U\rightarrow V$.
\end{problem}

\begin{problem}{2}
Sejam $U$ e $V$ espaços de Banach e $T\in\mathcal{L}(U,V)$. prove que se $T(U)$ é fechado em $V$ então $T(U)$ é isomorfo a $U/\ker(T)$.
\end{problem}

\begin{problem}{3}
    Sejam $U,V$ espaços de Banach e $T,T_1,T_2,\dots$, operadores em $\mathcal{L}(U,V)$ tais que $T_n(x)\to T(x)$ para todo $x\in U$. Mostre que para todo compacto $K\subset U$,
    $$\sup\limits_{x\in K}\Vert T_n(x)-T(x)\Vert\to 0.$$
\end{problem}

\begin{problem}{4}
Sejam $U$ um espaço de Banach real e $T:U\rightarrow U^{\ast}$ um operador linear tal que $T(x)(x)\geq 0$ para todo $x\in U$. Mostre que $T$ é contínuo.
\end{problem}

\begin{problem}{5}
Sejam $U$ um espaó de Banach, $V$ um espaço normado e $T:U\rightarrow V$ um operador linear de gráfico fechado. Mostre que se o operador inverso $T^{-1}$ existe e é contínuo, então a imagem de $T$ é um subespaço fechado de $V$.
\end{problem}

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\end{document}