0 like 0 dislike
72 views
[LINEAR FUNCTIONALS] In $R^{n}$ with the usual inner product, a linear functional $\ell:$ $\mathbb{R}^{n} \rightarrow \mathbb{R}$ is just a linear map into the reals (in a complex vector space, it maps into the complex numbers $\mathbb{C}$ ). Define the norm, $\|\ell\|$, as
$\|\ell\|:=\max _{\|x\|=1}|\ell(x)| .$
a) Show that the set of linear functionals with this norm is a normed linear space.
b) If $v \in \mathbb{R}^{n}$ is a given vector, define $\ell(x)=\langle x, v\rangle$. Show that $\ell$ is a linear functional and that $\|\ell\|=\|v\|$.
c) [REPRESENTATION OF A LINEAR FUNCTIONAL] Let $\ell$ be any linear functional. Show there is a unique vector $v \in \mathbb{R}^{n}$ so that $\ell(x):=\langle x, v\rangle$.
d) [EXTENSION OF A LINEAR FUNCTIONAL] Let $U \subset \mathbb{R}^{n}$ be a subspace of $\mathbb{R}^{n}$ and $\ell$ a linear functional defined on $U$ with norm $\|\ell\|_{U}$. Show there is a unique extension of $\ell$ to $\mathbb{R}^{n}$ with the property that $\|\ell\|_{\mathbb{R}^{n}}=\|\ell\|_{U}$.
[In other words define $\ell$ on all of $\mathbb{R}^{n}$ so that on $U$ this extended definition agrees with the original definition and so that its norm is unchanged].
| 72 views

0 like 0 dislike
0 like 0 dislike
0 like 0 dislike
0 like 0 dislike
0 like 0 dislike
0 like 0 dislike
0 like 0 dislike
1 like 0 dislike
0 like 0 dislike