Hermitian vector spaces

$\mathbb {C} O(n)=\left\{ A \in M_{n}(\mathbb {C}) \mid {}^{t} A \cdot A=E\right\}$ — the definition of the complex orthogonal group. It is clear that $O(n) \subset \mathbb {C} O(n)$ and $S O(n) \subset S \mathbb {C} O(n)$ (the subgroups of $O(n)$ and $\mathbb {C} O(n)$ consisting of elements with determinant 1). Can one, by analogy with Theorem 2, speak of the inclusions $\mathbb {C} O(n) \subset U(n)$ and $S \mathbb {C} O(n) \subset U(n)$?

Show that the metrics $d_{1}$ and $d_{2}$ from item 5 are not induced by any inner product on $\mathbb {R}^{n}$ (similarly on $\mathbb {C}^{n}$).

Using functional-analytic tools, verify that the formula

defines a norm on the space $\mathbb {R}^{n}$ for any $p \geqslant 1$ (the so-called $l_{p}$-norm). Verify that

Then one may formally regard $d_{1}(\mathbf{x}, \mathbf{y})=\left\| \mathbf{x}-\mathbf{y}\right\|_{\infty }$ (obviously $d_{2}(\mathbf{x}, \mathbf{y})=\left\| \mathbf{x}-\mathbf{y}\right\|_{1}, d(\mathbf{x}, \mathbf{y})=\left\| \mathbf{x}-\mathbf{y}\right\|_{2}$).

On the vector space $C(0,1)$ of continuous functions $f:[0,1] \rightarrow \mathbb {R}$ there exist analogues of the norms indicated above:

More generally: $\left\| f\right\|_{p}=\left(\int_{0}^{1}|f(t)|^{p} d t\right)^{1 / p}$ (this exercise is optional).