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\title{Math Studies Homework 3}
\author{James Cummings}

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    Submit the \LaTeX\ file containing your solutions to the
    email address algebrahw@legba.math.cmu.edu by
    1159 pm on Fri 18 Feb.
 


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\item  Halmos page 123 question 4 (don't miss out part e!)

\item  Recall that $l_2({\Bbb R})$  is the space of all infinite sequences $(x_n)$ such that
         $x_n \in {\Bbb R}$ and $\sum_n x_n^2 < \infty$, equipped with an inner product
       by writing $\langle x, y \rangle  = \sum_n x_n y_n$. 
       We defined the length or norm of $x$ to be $\| x \| = \langle x, x \rangle^{1/2}$ 
       and saw that if we set $d(x, y) = \| x - y \|$ then we get a metric. 

       Show that the ``unit ball'' $\{  x :  \| x \| \le 1 \}$ is not compact in this
       metric. 

\item  Show that the kernel of a continuous linear map from $l_2(\Bbb R)$ to $l_2(\Bbb R)$ is closed. 

\item  Find a linear map from $l_2(\Bbb R)$ to $l_2(\Bbb R)$ which is not continuous.

\item  Show that by contrast if we give ${\Bbb R}^n$ the usual Euclidean metric then all
       linear maps from ${\Bbb R}^n$ to ${\Bbb R}^n$ are continuous.

\item  Let $0 < q < p < r$, and let $a_i$ and $\mu_i$ be positive reals for $1 \le i \le n$.
  Show that 
\[
    \sum\limits_{i-1}^n \mu_i a_i^p \le \left( \sum_{i=1}^n \mu_i a_i^q \right)^{(r-p)/(r-q)} 
                                        \left( \sum_{i=1}^n \mu_i a_i^r \right)^{(p-q)/(r-q)} 
\]

      



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