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\centerline{\Large Math 675 Homework 7}
\centerline{Due 10/10/2018}
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\begin{enumerate}[1.]
\item Verify that the spaces $C[a,b], \ell_2, c, c_0, m$, and $\R^\infty$ are infinite-dimensional.
\item Let $f, f_1, \ldots, f_n$ be linear functionals on a linear space $L$ such that for all $x$ one has that $f_1(x)=f_2(x)=\ldots=f_n(x)=0$ implies $f(x)=0$. Prove that there are constants $a_1, \ldots, a_n$ such that $f=a_1f_1+\ldots+a_nf_n$.
\item Let $V$ be a vector space. Prove that the set of linear functionals on $V$ is a vector space. It is commonly denoted $V^*$.  Give an example of a non-trivial functional on $V^*$.
\item Explain the big-O and little-o notation for sequences in terms of quotient spaces.
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