The Riesz Representation Theorem

Prove Proposition 3.3: if $L$ is a bounded linear functional on a Hilbert space $\mathcal{H}$, then

Also, $\left|L(h)\right| \leq \left\| L\right\| \left\| h\right\|$ for every $h$ in $\mathcal{H}$.

Let $\mathcal{H} = l^2(\mathbb {N})$. If $N \geq 1$ and $L : \mathcal{H} \to \mathbb {F}$ is defined by $L(\left\{ \alpha_n\right\} ) = \alpha_N$, find the vector $h_0$ in $\mathcal{H}$ such that $L(h) = \left\langle h, h_0 \right\rangle$ for every $h$ in $\mathcal{H}$.

Let $\mathcal{H} = l^2(\mathbb {N} \cup \left\{ 0\right\} )$.

With the notation as in the preceding exercise, define $L : \mathcal{H} \to \mathbb {F}$ by $L(\left\{ \alpha_n\right\} ) = \sum_{n=1}^{\infty } n \alpha_n \lambda^{n-1}$, where $\left|\lambda \right| < 1$. Find a vector $h_0$ in $\mathcal{H}$ such that $L(h) = \left\langle h, h_0 \right\rangle$ for every $h$ in $\mathcal{H}$.

Let $\mathcal{H}$ be the Hilbert space described in Example 1.8 (the collection of all absolutely continuous functions $f : [0,1] \to \mathbb {F}$ with $f(0) = 0$ and $f' \in L^2(0,1)$, with $\left\langle f, g \right\rangle = \int_0^1 f'(t)\overline{g'(t)} \, dt$). If $0 < t \leq 1$, define $L : \mathcal{H} \to \mathbb {F}$ by $L(h) = h(t)$. Show that $L$ is a bounded linear functional, find $\left\| L\right\|$, and find the vector $h_0$ in $\mathcal{H}$ such that $L(h) = \left\langle h, h_0 \right\rangle$ for all $h$ in $\mathcal{H}$.

Let $\mathcal{H} = L^2(0,1)$ and let $C^{(1)}$ be the set of all continuous functions on $[0,1]$ that have a continuous derivative. Let $t \in [0,1]$ and define $L : C^{(1)} \to \mathbb {F}$ by $L(h) = h'(t)$. Show that there is no bounded linear functional on $\mathcal{H}$ that agrees with $L$ on $C^{(1)}$.