Functional Analysis_15. Inner Products and Orthogonality_15.1. Examples of Hilbert spaces

\begin{enumerate} \item[(i)] Let $\mathbb{R}^{n \times n}$ be the space of real $n \times n$ matrices and set $$\langle A, B \rangle := \operatorname{tr}(AB^{\!T}) \quad \text{for all } A, B \in \mathbb{R}^{n \times n},$$ where $\operatorname{tr}(M)$ is the trace of $M$ and $M^{\!T}$ is the transpose of $M \in \mathbb{R}^{n \times n}$. Show that this defines a Hilbert space. Deduce from this that $$|\operatorname{tr}(AB^{\!T})|^2 \le \operatorname{tr}(AA^{\!T})\,\operatorname{tr}(BB^{\!T}) \quad \text{for all } A,B\in\mathbb{R}^{n\times n}.$$ \pause \item[(ii)] Show that $(\ell^p, \|\cdot\|_p)$ is a Hilbert space if and only if $p = 2$. \pause \item[(iii)] Show that $(C([a, b]), \|\cdot\|_\infty)$ is not a Hilbert space. \end{enumerate}