Exercises 4.9 Exercises
1. Closed nullspace (level 1).
Let \(X\) be a Banach space and \(A:X\to X\) a bounded linear map. Show that the null-space ker\((A)\) is closed.2. Closed range (level 1).
Let \(X\) be a Banach space and \(A:X\to X\) a bounded linear map. Assume there is a \(K>0\) such that
\begin{equation*}
\|x\|\leq K \|Ax\|, \qquad \mbox{for all} \qquad x\in X.
\end{equation*}
Show that the range of \(A\) is closed. 3. Spectral representation (level 2).
Let \(A\) be a symmetric linear operator on a Hilbert space \(H\) with \(R(A)=H\) and with compact inverse \(A^{-1}\text{.}\) The natural domain of definition is
\begin{equation*}
{\cal D}(A) = \{ u\in H; Au\in H\}.
\end{equation*}
Show that there exists an orthonormal basis \(\{w_j\}\) of \(H\) such that
\begin{equation*}
{\cal D}(A) = \left\{ u; u=\sum c_j w_j,\qquad \sum |c_j|^2\lambda_j^2<\infty\right\}.
\end{equation*}
4. Normal Integral Operators (level 2).
For \(k\in L^2(\Omega\times\Omega)\) consider the integral operator
\begin{equation*}
u \mapsto Ku(x) := \int_\Omega k(x,y) u(y) dy .
\end{equation*}
- Find a condition on the kernel \(k\) such that the integral operator \(K\) is normal.
- Find an example of a normal integral operator that is not symmetric. Make sure to chose \(k\) and \(\Omega\) so that \(k\in L^2(\Omega\times \Omega)\text{.}\)
5. Root of Laplacian (level 1).
For \(A=-\Delta\) show that on a bounded domain the norm on \({\cal D}(A^{\frac{1}{2}})\) is equivalent to the norm on \(H_0^1\text{.}\) (Hint: If \(A\) is symmetric, \((Au,v)=(u,Av)\text{,}\) then also \(A^\frac{1}{2}\)).6. Computation of operator norms (level 1).
Evaluate the norms of the following linear operators on \(C^0([0,1])\text{:}\)- \(\displaystyle L_1:f(x)\mapsto \int_{[0,x]}f(s)ds;\)
- \(\displaystyle L_2:f(x)\mapsto x^2f(0);\)
- \(\displaystyle L_3:f(x)\mapsto f(x^2);\)
- \(\displaystyle L_4=L_1^n, n\geq1.\)
7. Ptolemaic inequality (level 3).
Let \(H\) be a Hilbert space with norm \(\|\cdot\|\) and scalar product \(\langle\cdot,\cdot\rangle\text{.}\)- Let \(x,y\in H\setminus\{0\}\) and set\begin{equation*} \tilde x=\frac{x}{\|x\|^2}, \tilde y=\frac{y}{\|y\|^2}. \end{equation*}Show that\begin{equation*} \|\tilde x-\tilde y\| = \frac{\|x-y\|}{\|x\|\|y\|}. \end{equation*}Hint: use the fact that the norm comes from an inner product.
- Show that, for any \(x,y,z,w\in H\) holds the Ptolemaic inequality:\begin{equation*} \|x-z\|\,\|y-w\|\leq\|x-y\|\,\|z-w\|+\|y-z\|\,\|x-w\|. \end{equation*}Hint: show first that \(\langle a-b,c-d\rangle+\langle a-d,b-c\rangle=\langle a-c,b-d\rangle\text{.}\)
- \(B\subset H\)\(tx+(1-t)y\in B\)\(x,y\in B\)\(t\in[0,1]\)\(x,y\in\partial B\text{,}\)\(tx+(1-t)y\in \partial B\)\(t=0,1\text{.}\) Hint:
8. Norms on \(C^1([0,1])\) (level 1).
Which of these functions are norms on \(C^1([0,1])\text{?}\)- \(\displaystyle N_1(f)=\|f'(x)\|_{C^0};\)
- \(\displaystyle N_2(f)=|f(0)-f(1)|+\|f'(x)\|_{C^0};\)
- \(\displaystyle N_3(f)=|f(0)|+\|f'(x)\|_{C^0};\)
- \(\displaystyle N_4(f)=\|f\|_{L^1}+\|f'(x)\|_{C^0}.\)
9. Convolution (level 2).
Let \(g\in L^1(\bR)\) be non-negative and let \(p\geq1\text{.}\) Show that the linear map \(L:L^p(\bR)\to L^p(\bR)\) defined by \(Lf=g*f\) (convolution of \(g\) with \(f\)) is bounded and \(\|L\|_{op}=\|g\|_{L^1}\text{.}\)10. Distance from a set (level 1).
Let
\begin{equation*}
S = \left\{f\in L^p([0,1])\,:\,\int_{[0,1]}f^2(s)ds=1\right\}.
\end{equation*}
Find the \(L^p\)-distance of \(f(x)=x\) from \(S\) for \(p=1,2,\infty\text{.}\)