Elementary Properties of the Scalar Product

Prove that the requirement of bilinearity (sesquilinearity) in the axioms of a Euclidean (Hermitian) space can be replaced by linearity in the second argument.

Show that the function $f$ of the vectors $x$ and $y$ of the real plane, given by the formula

defines a Euclidean inner product on the plane. Find the inner product of the vectors $(1,-1)$ and $(4,1)$.

Consider the function $f$ of the vectors $x$ and $y$ of a two-dimensional space, defined as follows:

Prove that this formula defines a Euclidean inner product if and only if $a > 0, a d-b^{2} > 0$.

Find the lengths of the following vectors of the Hermitian space $\mathbb {C}^{3}$:

Check whether the following bilinear functions on the space of polynomials of degree at most $n$ are Euclidean inner products:

The polynomials $x^{2}$ and $x^{4}$ are given. Find their lengths and the angle $\varphi$ between them in the Euclidean space of polynomials of degree at most 4 with the inner product given by the integral:

Check whether:

Using the Cauchy--Bunyakovsky--Schwarz inequality, prove the following inequalities:

Using the Cauchy--Bunyakovsky--Schwarz inequality, prove that the triangle inequalities

hold in a Euclidean (Hermitian) space, where the right-hand inequality becomes an equality if and only if the vectors $x$ and $y$ are collinear and equally directed.

(Pythagorean theorem). Prove that if the vectors $x$ and $y$ in a Euclidean (Hermitian) space are orthogonal, then

Is the converse statement true?

Prove that in the parallelogram spanned by the vectors $x$ and $y$ of a Euclidean space, the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the sides. Is the analogous statement true for a Hermitian space?

Prove that every subspace $L$ of a Euclidean (Hermitian) space $V$ is a Euclidean (Hermitian) space with respect to the restriction to $L$ of the inner product from $V$, i.e., the inner product defined as follows: the inner product $\left(l_{1}, l_{2}\right)_{L}$ of two vectors $l_{1}, l_{2} \in L$ equals their inner product $\left(l_{1}, l_{2}\right)_{V}$ as vectors of $V$. Such an inner product is called the inner product induced by the embedding $L \subset V$ of the subspace $L$ in the space $V$.

On the external direct sum $V \oplus W$ of two Euclidean (Hermitian) spaces $V$ and $W$, introduce a Euclidean (Hermitian) inner product so that the inner products on $V$ and $W$ induced by the standard embeddings

coincide with their original inner products $(\cdot , \cdot )_{V}$ and $(\cdot , \cdot )_{W}$ respectively, and so that the subspaces $V$ and $W$ are orthogonal to each other.

Prove the Cauchy--Bunyakovsky--Schwarz inequality for a Hermitian space:

Let $V$ be a real Euclidean space with inner product $(\cdot , \cdot )_{\mathbb {R}}$. Show that its complexification $V_{\mathbb {C}}$ can be turned into a Hermitian space by introducing on it the Hermitian inner product $(\cdot , \cdot )_{\mathbb {C}}$ by the formula

The space $V_{\mathbb {C}}$ with the Hermitian product $(\cdot , \cdot )_{\mathbb {C}}$ defined in this way will be called the complexification $V_{\mathbb {C}}$ of the Euclidean space $V$. Which Hermitian space is obtained by complexifying the standard $\mathbb {R}^{n}$?

Prove that if $e_{1}, e_{2}, \ldots , e_{n}$ is an orthonormal basis in a real Euclidean space $V$, then the vectors $e_{1}+i \cdot 0$, $e_{2}+i \cdot 0, \ldots , e_{n}+i \cdot 0$ form an orthonormal basis in the complexification $V_{\mathbb {C}}$ with respect to the Hermitian inner product introduced in the previous problem.

Let $W$ be a Hermitian space with Hermitian inner product $(\cdot ,\cdot )$. Consider its realification $W_{\mathbb {R}}$. Prove that $W_{\mathbb {R}}$ becomes a Euclidean space if one introduces on $W_{\mathbb {R}}$ the Euclidean inner product $(\cdot , \cdot )_{\mathbb {R}}$ by the formula

In the case $W = \mathbb {C}^{n}$, write the formula for the value $\left(z_{1}, z_{2}\right)_{\mathbb {R}}$ in the standard coordinates $\left(x^{1}, \ldots , x^{n}, y^{1}, \ldots , y^{n}\right)$ of the realified space $W_{\mathbb {R}}$, given by the equalities $x^{k}+i y^{k} = z^{k}$. Show that if the standard Hermitian space $\mathbb {C}^{n}$ is realified in this way, the result is the standard Euclidean space $\mathbb {R}^{2 n}$.

Show that, under the conditions of the previous problem, the vectors $z$ and $i z$ are always orthogonal with respect to $(\cdot , \cdot )_{\mathbb {R}}$.

Problem 1284: $W$ is a Hermitian space with Hermitian inner product $(\cdot , \cdot )$, $W_{\mathbb {R}}$ is its realification with the Euclidean inner product $\left(z_{1}, z_{2}\right)_{\mathbb {R}} = \operatorname {Re}\left(z_{1}, z_{2}\right)$.

Show that the imaginary part $\operatorname {Im}(\cdot , \cdot )$ of a Hermitian inner product is a real skew-symmetric bilinear function on the Hermitian space $W$, and that the form $\operatorname {Im}(\cdot , \cdot )$ is invariant under multiplication by $i$, i.e.,

Show that the Hermitian inner product $(\cdot , \cdot )$ is uniquely recovered both from its real part $\operatorname {Re}(\cdot , \cdot )$ and from its imaginary part $\operatorname {Im}(\cdot , \cdot )$. To this end, express: