Convex Sets and Projection Theorems

Let \(C\) be a closed convex set in a Hilbert space \(H\). Show that \(C\) contains a unique element of minimal norm.

Let \(X\) be a real Banach space and \(X^*\) denote its dual. Let \(C \subset X\) be a convex set containing the origin. Define

The goal of this problem is to prove that \(C^{**} = \overline {C}\).

1. Show that \(C^{**}\) is closed.
2. Show that \(\overline {C} \subset C^{**}\).
3. Show that \(\overline {C}\) is convex.

4. Assume there is \(x_0 \in X\) such that \(x_0 \in C^{**}\) but \(x_0 \notin \overline {C}\). Show there exists an \(f \in X^*\) such that for all \(x \in \overline {C}\) we have:

   \[f(x) < 1 < f(x_0).\]
5. Explain the contradiction in (d) and how it can be used to conclude with the desired result.

Let \(X\) be a normed vector space. Let \(x_0 \in X\) and consider the closed ball \(\widetilde {B}(0,\|x_0\|)\). Use the Hahn-Banach theorem to show that there exists \(f \in X'\) such that \(f(x_0) = 1\) and for any \(x \in \widetilde {B}(0,\|x_0\|)\) we have \(f(x) \le 1\). Do not use Hahn-Banach for convex sets or the separating hyperplane theorem.

Suppose that \(\rho (x,t)\) is the number density of cars evolving according to the traffic model \[ \frac {\partial \rho }{\partial t}+\frac {\partial (\rho u)}{\partial x}=0. \] with \(u\) the car speed. Take \(u=1-\rho \). A queue is building up at a traffic light \(x=1\) so that, when the light turns green at \(t=0\), \[ \rho (x,0)= \begin {cases} 0, & x<0\text { and }x>1,\\ x, & 0<x<1. \end {cases} \]

1. Use the method of characteristics to solve the Cauchy problem. In particular, construct the characteristic projections and identify any shocks or rarefaction waves.
2. Hence, show that a collision first occurs at \(x=1/2\) when \(t=1/2\) and, thereafter, there is a shock \(S(t)\). Use the Rankine-Hugoniot condition to derive the following expression for the speed of the shock: \[ \frac {dS}{dt}=\frac {S+t-1}{2t}. \]

Consider the scalar linear equation \[ a\frac {\partial u}{\partial x}+b\frac {\partial u}{\partial y}=\alpha u \] with \(a,b,\alpha \) differentiable functions of \(x,y\). Suppose that \(u\) is prescribed on some arc \(\Gamma \) (Cauchy data).

1. By introducing a test function \(\psi \) that vanishes on an arbitrary curve \(\gamma \) and applying Green’s theorem to the domain \(D\) bounded by \(\partial D=\Gamma \cup \gamma \), derive the integral equation for a weak solution to the Cauchy problem.
2. Extend the analysis to the case where \(u\) is discontinuous across an open curve \(C_0\) within the domain \(D\). Hence show that \(C_0\) is a characteristic projection.
3. Write down the generalization of part (b) to the case of the quaslinear equation \[ \frac {\partial P}{\partial x}+\frac {\partial Q}{\partial y}=c \] where \(P,Q,c\) are differentiable functions of \(x,y,u\), and hence derive the Rankine–Hugonoit condition.

Let \(C\) be a closed convex set in a Hilbert space \(H\), and let \(u\) be an arbitrary vector in \(H\). Show that there exists one and only one element in \(C\) which is closest to \(u\).

Consider \[ \begin {aligned} \frac {dx}{dt} &= Ax+f(x),\\ x(0) &= x_0 \end {aligned} \tag {2.1} \] where \(x\in \mathbb {R}^n\) and \(A\) is a constant \(n\times n\) matrix. Suppose that \(A\) is semisimple, that \(f\in C^2\) and that \(f(0)=0\), \(D_xf(0)=0\).

1. Define what is meant by the stable manifold, the center manifold and the unstable manifold at the origin for (2.1).
2. Give a representation of the solution of (2.1) when \(f=0\), in terms of the eigenvalues of \(A\) and the associated projections and nilpotents. Indicate how to compute the projections.
3. Discuss the asymptotic behavior of the solution in (b) for \(t\to \infty \). Characterize the manifolds defined in (a) as precisely as possible.
4. Give a representation of the solution of (2.1) for \(f\ne 0\). Under what conditions, if any, on \(A\) and the initial data can one conclude that this solution exists for all \(t\in \mathbb {R}^+\)? You need not prove all the assertions for the last part, but should at least state your reasons.
5. State the Hartman-Grobman theorem for \[ x'=F(x) \] and state clearly what further conditions, if any, are needed on (2.1) to apply the Hartman-Grobman theorem to (2.1).

A bounded linear operator \(P\) in a complex Hilbert space \(\mathcal {H}\) is a projection if and only if \(P^2=P\). Prove that a projection \(P\) is an orthogonal projection (so that \(P\mathcal {H}\perp (1-P)\mathcal {H}\)) if and only if \(P\) is selfadjoint. (You may use the known theorem that a bounded linear operator \(A\) in \(\mathcal {H}\) is selfadjoint if and only if \((Ax,x)\) is real for all \(x\in \mathcal {H}\).)

Let \(\mathcal {H}\) be a Hilbert space with inner product \((x,y)\) and let the set \(\{e_n\}_{n=1}^{\infty }\) be orthonormal in \(\mathcal {H}\). Complete the following:

1. Prove that the formal series \[ g=\sum _{n=1}^{\infty }(f,e_n)e_n \] converges for each \(f\in \mathcal {H}\) to an element \(g\in \mathcal {H}\).
2. Let the mapping \(\mathcal {P}\) be defined on \(\mathcal {H}\) by \(\mathcal {P}f=g\), \(g=\sum _{n=1}^{\infty }(f,e_n)e_n\). Prove that \(\mathcal {P}\) is an orthogonal projection on \(\mathcal {H}\).