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\centerline{\Large Math 675 Homework 11}
\centerline{Due 11/28/2018}
\vspace{.25in}

\begin{enumerate}[1.]
\item Let $V$ be a normed real vector space and $F: V\rightarrow \R$ a linear functional. Prove that $F$ is continuous if and only if $N(F)$ is closed. (Hint: The forward direction requires no work at all; for the backward direction, assume $F$ is not bounded and show that $N(F)$ is not closed by perturbing a non-zero point.)
\item Prove that each of the following functionals is continuous on $C_\infty[0,1]$ and compute its norm:
\begin{enumerate}
\item $f(x)=ax(0)+bx(1)$,
\item $g(x)=\int_0^{1/2}x(t)dt - \int_{1/2}^1 x(t)dt$
\end{enumerate}
\item Prove that if $p<q$ and $f$ is a linear functional on $C_p[a,b]$ then it is also continuous on $C_q[a,b]$.
\item Give an example (with justification) of a linear functionals $f$ and $g$ on $C[a,b]$ such that:
\begin{enumerate}
\item $f$ is continuous with respect to $d_\infty$ but not $d_1$.
\item $g$ is continuous with respect to $d_2$ but not $d_1$.
\end{enumerate}
\item Let $V_0$ be a normed real vector space and $V$ its completion. Prove that $V_0^*$ and $V^*$ are  isomorphic  Banach spaces.
\end{enumerate}





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