Inst. för Matematik    |   KTH    |



5B1472 Funktionalanalys, vt 2004

Hemuppgifter 2


1. Let $X$ be a Banach space over the reals or the complex numbers. Suppose there exists a sequence $x_1,x_2,x_3,\ldots$ in $X$ such that the linear span of these vectors is dense in $X$. Show that $X$ is separable, which means that there exists some other sequence $y_1,y_2,y_3,\ldots$ which is dense in $X$.

2.Let $X$ be a Banach space and $Y$ a linear subspace of $X$ (not necessarily closed). Show that -- up to equivalence of norms -- there is only one way to supply $Y$ with a norm so that (1) $Y$ becomes a Banach space and (2) the injection mapping $Y\to X$ is continuous.

3. Let $T$ be a mapping from the reals $R$ to the reals $R$, which is additive: $T(x_1+x_2)=T(x_1)+T(x_2)$. Prove that if $T$ is continuous, then $T$ is linear: $T(x)=\alpha\,x$, for some real $\alpha$. Try to decide whether the conclusion remains valid if we drop the assumption of continuity.