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Hemuppgifter 31. Let $X$ be a Banach space over the complex numbers. Let a sequence of vectors $x_1,x_2,x_3,\ldots$ be given in $X$. Show that the linear span of these vectors $x_1,x_2,x_3,\ldots$ is dense if and only if the following criterion is met: Whenever $\phi$ is a bounded linear functional on $X$, and $\phi(x_1)=\phi(x_2)=\phi(x_3)=\ldots=0$, then $\phi=0$. 2. Let $C[0,1]$ be the space of continuous functions on the interval $[0,1]$, with the supremum norm, as usual. Show that $C[0,1]$ is separable, while its dual space $C[0,1]^*$ is not. Conclude that $C[0,1]$ is not a reflexive space. 3. Let $H$ be the Hilbert space (over the complex field) of analytic functions which are $L^2$ with respect to area measure on the unit disk. Show that for each point $w_0$ in the open unit disk, the linear functional $f\mapsto f(w_0)$ is continuous on $H$. Then -- by Riesz' theorem -- there exists an element $k_{w_0}$ of the space which gives this functional as the inner product with $k_{w_0}$. Find this function $k_{w_0}$. It is called the Bergman kernel. |