Due date: Monday 07. October 2024, 10 AM.
Exercise 1
Let \((V,\langle \cdot{,}\cdot\rangle)\) be a Euclidean space and \(u,v\in V.\) Show that \(\langle u,v\rangle=0\) if and only if \[\|u\| \leqslant \|u + s v\|\] for all \(s\in\mathbb{R}.\)
Exercise 2
Let \(V=\mathsf C([-1,1],\mathbb{R})\) be the \(\mathbb{R}\)-vector space of continuous real-valued functions defined on the interval \([-1,1]\) and let \(U=\operatorname{span}\{f,g,h\}\subset V\) be the subspace generated by \(f(x) = 1, g(x)=x\) and \(h(x)=x^2.\) Let furthermore \(\langle \cdot{,}\cdot\rangle:V\times V\to \mathbb{R}\) be given by \[\langle f,g\rangle =\int_{-1}^{1}f(x)g(x)\mathrm dx.\] Compute \(\Pi^\perp_U(a),\) where \(a(x) = |x|\) for all \(x\in [-1,1]\) and \(\Pi^\perp_U:V\to V\) is the orthogonal projection onto \(U.\)
Exercise 3
Let \(V\) be an \(\mathbb{R}\)-vector space.
Show that if \(\langle \cdot{,}\cdot\rangle:V\times V\to \mathbb{R}\) is an inner product on \(V,\) then the induced norm \(\|\cdot\|:V\to \mathbb{R}\) satisfies the parallelogram law \[\|v+w\|^2+\|v-w\|^2 = 2(\|v\|^2+\|w\|^2)\] for all \(v,w\in V.\)
Show that if \(V=\mathbb{R}^2,\) the norm \(\|\cdot\| :V\to \mathbb{R}\) defined by \[\|\vec v\| = \left\|\begin{pmatrix} v_1\\ v_2\end{pmatrix}\right\| = |v_1|+|v_2|\] cannot be induced by an inner product \(\langle \cdot{,}\cdot\rangle:V\times V\to\mathbb{R}.\)
Exercise 4
Let \((V,\langle \cdot{,}\cdot\rangle)\) be a Euclidean space and \(U,W\subset V\) be subspaces. Show that \(\Pi^\perp_U\circ \Pi^\perp_W:V\to V\) is the zero map if and only if \(\langle u,w\rangle = 0\) for all \(u\in U, w\in W.\)