Definition (27.1-27.9)

Will a complex two-dimensional vector space be unitary if the scalar product in it is defined by the following function of the coordinates of the vectors:

The unitary norm of a matrix is defined as its length under the scalar product defined in problem 27.2(1). Prove that the unitary norm equals the square root of the sum of the squares of the moduli of all elements of the matrix.

Let $e$ be a basis in a complex vector space $\mathcal{E}$. Prove that in $\mathcal{E}$ there exists one and only one unitary scalar product with respect to which the basis $e$ is orthonormal.

Let two unitary scalar products $(x, y)_{1}$ and $(x, y)_{2}$ be given in a complex vector space. Prove that for any positive real numbers $\lambda$ and $\mu$ the function $(x, y) = \lambda (x, y)_{1} + \mu (x, y)_{2}$ is also a unitary scalar product.

Prove that in a unitary space the identities from problem 25. 13 do not hold for all pairs of vectors.

Let two scalar products $(x, y)_{1} = (x, y)_{2}$ be given in a complex vector space, and let every vector have equal lengths in each of them: $(x, x)_{1} = (x, x)_{2}$. Prove that the scalar products coincide.

Prove that a triangle with sides $x, y$ and $z$ in a unitary space is right-angled if $\left|z\right| = \left|x\right| \cos (\widehat{x, z})$, and may fail to be right-angled if $\left|z\right| = \left|x\right| \cos (\widehat{z, x})$.

Prove that in a unitary space $(x, y) = 0$ implies $\left|x\right|^{2} + \left|y\right|^{2} = \left|x + y\right|^{2}$. What can be said about the product $(x, y)$ if the latter equality holds?