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{\Large 18.099 - 18.06CI.}
\vspace{1cm}
{\large Attention: Next week at MIT Tuesday is a virtual Monday; and
Monday is a holiday.}
{Due on Tuesday, Feb 17 in class.
First draft due on Thursday, Feb 12}.
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\begin{enumerate}
\item{Give an example of a linear space over the rationals which is
not a linear space over the reals. Can you find an example of a real
linear space which is not a rational linear space? Explain your answer.}
\item{Let $A$ be the linear space of real infinite sequences
$(a_1, a_2, a_3, \ldots)$
over ${\mathbb R}$ with coordinatewise addition and multiplication by
numbers. Which of the following
are linear subspaces in $A$?
\begin{enumerate}
\item{Sequences with only finitely many nonzero terms;}
\item{Sequences with only finitely many zero terms;}
\item{Cauchy sequences, namely $\{a_i\}_{i=1}^\infty$ satisfying the
following condition:
for every $\varepsilon$ there exists
a number $N>0$ such that $|a_n -a_m| < \varepsilon $ for all $m,n >N$;}
\item{Sequences $\{a_i\}_{i=1}^\infty$ for which
the series $\sum_{i=1}^\infty a_i^2$ converges.}
\end{enumerate}
Prove your answers.}
\end{enumerate}
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