Math 6210 • Real Analysis

The space \(C[0,1]\) of continuous functions on \([0,1]\) is a subspace
of all the \(L^p\) spaces, \(L^p[0,1]\) for \(1\le p\le\infty\). Define a linear
operator \(T:C[0,1]\to\mathbb R\) by \(Tf=f(1/2)\). Show that \(T\) is a bounded
operator for the \(L^p\)-norm if and only if \(p=\infty\). Show that \(T\) extends to a
bounded operator on all of \(L^\infty[0,1]\).

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\textbf{B. Complex Variables}

Do at least 3 of these.
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In what follows, \(D_R(z_0)\) and \(\overline{D_R}(z_0)\) denote the open and closed
discs, respectively, centered at \(z_0\), with radius \(R\).