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3. If \(d\) is a metric on \(M\), show that
\[|d(x,z)-d(y,z)|\leq d(x,y),\]
for any \(x,y,z\in M\).
4. Let \(f:[0,1]\to\mathbb{R}\) be continuously differentiable, with \(f(0)=0\). Prove that
\[\sup_{0\leq x\leq 1}|f(x)|\leq\sqrt{\int_{0}^{1}(f^{\prime}(x))^{2}\,dx}.\]
Hint: Apply CSB inequality to \(|f(x)|=|\int_{0}^{x}f^{\prime}(t)dt|\).
5. Prove that in an inner product space we have additivity in the second slot as well as the first slot.
6. Prove that in an inner product space we have conjugate homogeneity in the second slot, i.e., prove that
\[\langle u,\alpha v\rangle=\overline{\alpha}\langle u,v\rangle,\]
for all scalars \(\alpha\).
7. Let \(u\) and \(v\) be vectors in an inner product space \(V\). Give an alternate proof of the CSB inequality by answering the following questions:
1. Let \(u\neq 0\) and \(t\) be a scalar. Then consider \[\langle tu+v,tu+b\rangle\geq 0,\] for all values of \(t\). Expand this inequality to obtain a quadratic inequality of the form \[at^{2}+bt+c\geq 0.\] What are \(a,b\), and \(c\) in terms of \(u\) and \(v\)?
2. Use your knowledge of quadratic equations and their graphs to obtain a condition on \(a,b\), and \(c\) for which the inequality in part a) is true.
* Show that in terms of \(u\) and \(v\), your condition in part b) is equivalent to the CSB inequality.
8. Show that
\[||x||_{\infty}\leq||x||_{2}\leq||x||_{1},\]
for any \(x\in\mathbb{R}^{n}\). Also show the following:
\[||x||_{1} \leq n||x||_{\infty},\] \[||x||_{1} \leq \sqrt{n}||x||_{2}.\]
9. Show that if \(A=\{a_{ij}\}\) is an \(m\times n\) matrix then
\[||A||=\max_{1\leq i\leq n}\left(\sum_{j=1}^{m}|a_{ij}|^{2}\right)^{1/2}\]
is a norm on the vector space \(M_{m\times n}\) of all \(m\times n\) matrices.
10. Let \(A\) and \(B\) be \(n\times n\) complex matrices
* Show that \(\langle A,B\rangle=\mathrm{tr}(AB^{*})\) defines an inner product on the space of \(n\times n\) complex matrices.
* Prove that \(|\mathrm{tr}(AB^{*})|^{2}\leq\mathrm{tr}(AA^{*})\mathrm{tr}(BB^{*})\).
11. In a normed space, prove that
* \(||x-y||\geq|||x||-||y|||\),
* \(||(1/\lambda)x||=1\) if \(\lambda=||x||\), \(x\neq 0\).
12. Let \(\{x_{n}\}\) be a sequence in a normed space and suppose that \(\sum_{k=1}^{\infty}(x_{k}-x_{k+1})\) is absolutely convergent. Determine whether \(\{x_{n}\}\) is Cauchy or convergent.
13. Let \(H\) be an inner product space with
\[\langle f,g\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}f(t)\overline{g(t)}dt.\]
* Show that \[S=\{f_{n}(t)=e^{int}:n\,\mathrm{is\ an\ integer}\,\}\] is on orthonormal set. (Recall that \(e^{int}=\mathrm{cos}nt+i\,\mathrm{sin}nt,\ 0\leq t\leq 2\pi\) and \(\overline{e^{int}}=e^{-int}\))* Let \(W=\operatorname{span}(S)\) and let \(T\) and \(U\) be operators defined on \(W\) by \[T(f_{n})=f_{n+1}\,\text{ and }\,U(f_{n})=f_{n-1}.\] Show that \(U=T^{*}\). (Recall that \(U=T^{*}\) if \(\langle Tf_{n},f_{m}\rangle=\langle f_{n},Uf_{m}\rangle\)).
14. Suppose \(C[-1,1]\) is the vector space of real-valued continuous functions on \([-1,1]\) with the inner product giving by
\[\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)\,dx\]
for \(f,g\in C[-1,1]\). Let \(\phi\) be the linear functional on \(C[-1,1]\) defined by \(\phi(f)=f(0)\). Show that \(\phi(f)=f(0)\). Show that there does **not** exist \(g\in C[-1,1]\)such that
\[\phi(f)=\langle f,g\rangle\]
for every \(f\in C[-1,1]\). Is this a contradiction to Riesz representation theorem?
15. The _diameter_ of a set of a nonempty set \(A\) of a metric space \((M,d)\) is defined by
\[\operatorname{diam}(A)=\sup\{d(a,b):a,b\in A\}.\]
Show that \(A\) is bounded if and only if \(\operatorname{diam}(A)\) is finite.
### 2.2 Fixed Point Theorems and Applications
One of the most common problems in mathematics is the following:
Given a linear mapping \(T\) and the image \(y\), solve \(Tx=y\) for \(x\).
If the mapping \(T:\mathbb{R}^{n}\to\mathbb{R}^{n}\), then this problem becomes that of solving a set of simultaneous linear equations. We often encounter the need to solve \(f(x)=0\), where \(f\) is a real-valued function. Of course when \(f\) is a linear or a quadratic function, we know how to solve it. But for most other functions we use some method of approximating the roots. For example _Newton’s method_ provides one such approximation under certain conditions. We assume \(x_{0}\) to be an approximation to the root and then calculate \(x_{1}=x_{0}-\dfrac{f(x_{0})}{f^{\prime}(x_{0})}\); next we will use \(x_{1}\) in this formula to obtain a better approximation \(x_{2}\) and so on. Such a process is called _iterative_. Our hope is to get a sequence of points \(x_{0},x_{1},\ldots\) such that this sequence converges to the root of \(f(x)=0\).
Completeness is a useful property if one is interested in solving equations. For example how does one compute \(\sqrt{2}\) by hand? You would most likely start by finding an approximate solution to the equation \(x^{2}=2\) and then seek ways to improve the estimate. Better and better estimates give you a sequence of points,but how do we know this sequence converges? The completeness of \(\mathbb{R}\) comes to the rescue, since the sequence converges to a point in \(\mathbb{R}\). The modern metric space version of the method of successive approximation was first explicitly stated in Banach’s thesis in 1922. He considered a complete metric space \(M\) and a contraction mapping \(f:M\to M\) and showed how to get the unique solution to \(f(x)=x\). This theorem is now referred as the Banach Contraction Mapping Theorem which we will explain in detail below. However, we first examine a continuous function \(f:[-1,1]\to[-1,1]\) shown in Figure 2.9.
Figure 2.9 indicates that the graph of \(f\) must cross the diagonal line; more precisely, there must be a point \(x_{0}\in[-1,1]\) such that \(f(x_{0})=x_{0}\). This is easy to see if we first set up the function
\[F(x)=f(x)-x.\]
Observe that \(F\) is defined on \([-1,1]\) and \(F\), being the difference of two continuous functions, is continuous on \([-1,1]\). Furthermore, \(F(1)\geq 0\) and \(F(-1)\leq 0\). Therefore, we can apply the Intermediate Value Theorem to conclude that there exists a point \(x_{0}\) in \([-1,1]\) such that \(F(x_{0})=0\) or equivalently \(f(x_{0})=x_{0}\).
**Definition 69** (Fixed Point).: A **fixed point** of a mapping \(f:A\to A\) is an \(x\in A\) which is mapped onto itself (is “kept fixed” by \(f\)), that is,
\[f(x)=x;\]
the image \(f(x)\) coincides with \(x\).
Given a mapping there is no guarantee that it has a fixed point, let alone that it has a unique fixed point. For example, a translation \(f:\mathbb{R}\to\mathbb{R}\), \(f(x)=x+a\) where \(a\neq 0\), has no fixed points. On the other hand, \(f:\mathbb{R}\to\mathbb{R}\) where \(f(x)=x^{2}\)has two fixed points \(0\) and \(1\) as we can see \(f(0)=0\) and \(f(1)=1\). The mapping (projection) \(P\) of \(\mathbb{R}^{2}\) onto the \(x\)-axis where \(P(x,y)=(x,0)\) has infinitely many fixed points, namely the entire \(x\)-axis where points are of the form \((x,0)\).
**Definition 70** (Contraction).: Let \(X=(X,d)\) be a metric space. A mapping \(T:X\to X\) is called a **contraction on \(X\)** if there is a positive constant \(\alpha<1\) such that for all \(x,y\in X\)
\[d(Tx,Ty)\leq\alpha\ d(x,y).\]
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