Main concepts and properties. Examples of Euclidean and Hilbert spaces

Prove the Cauchy-Bunyakovsky inequality. Prove that the equality $\left|(x, y)\right| = \left\| x\right\| \cdot \left\| y\right\|$ holds if and only if the vectors $x$ and $y$ are linearly dependent.

Prove that the function defined by equality (3.1) satisfies all the axioms of a norm.

Let $E$ be a Euclidean space. Prove the continuity of the inner product in each argument and jointly in both arguments.

Prove the Pythagorean theorem in a Euclidean space $E$: $x \perp y$ if and only if $\left\| x+y\right\|^{2} = \left\| x\right\|^{2}+\left\| y\right\|^{2}$. In a real Euclidean space prove the law of cosines: for any $x, y \in E$ it holds that

Let $\left\{ x_{n}\right\}_{1}^{\infty }$ and $\left\{ y_{n}\right\}_{1}^{\infty }$ be sequences of vectors of a Euclidean space, with $\left\| x_{n}\right\| \leqslant 1,\left\| y_{n}\right\| \leqslant 1$ for all $n \in \mathbb {N}$. Prove that if $\left(x_{n}, y_{n}\right) \rightarrow 1$, then $\left\| x_{n}-y_{n}\right\| \rightarrow 0$.

Let $E$ be a Euclidean space. Prove the polarization identity: for any vectors $x$ and $y$ from $E$ the equality $(x, y) = \frac{1}{4} \sum_{k = 0}^{3} i^{k}\left\| x+i^{k} y\right\|^{2}$ holds in the case of a complex space $E$, or the equality $(x, y) = \frac{1}{4}\left\| x+y\right\|^{2}-\frac{1}{4}\left\| x-y\right\|^{2}$ holds in the case of a real space $E$.

Let $\left(H_{1},(\cdot , \cdot )_{H_{1}}\right)$ and $(H_{2},(\cdot , \cdot )_{H_{2}})$ be Hilbert spaces. Denote by $\left\| \cdot \right\|_{H_{1}}$ and $\left\| \cdot \right\|_{H_{2}}$ the norms generated by the inner products. Prove that the Hilbert spaces $\left(H_{1},(\cdot , \cdot )_{H_{1}}\right)$ and $\left(H_{2},(\cdot , \cdot )_{H_{2}}\right)$ are unitarily equivalent if and only if the Banach spaces $(H_{1},\left\| \cdot \right\|_{H_{1}})$ and $(H_{2},\left\| \cdot \right\|_{H_{2}})$ are isometrically isomorphic (see Definition 2.2).

Definition 2.2: A linear mapping $J$ of a normed space $X$ into a normed space $Y$ is called an isometric embedding (isometry) if $J$ preserves the norm, i.e., $\left\| J(x)\right\|_{Y} = \left\| x\right\|_{X}$; an isometric isomorphism if $J$ is a bijective isometric embedding; an embedding if $J$ is continuous and injective; an isomorphism if $J$ is a bijective embedding and the inverse mapping $J^{-1}$ is continuous. Two normed spaces between which an isomorphism (isometric isomorphism) can be constructed are called isomorphic (respectively, isometrically isomorphic).

Prove Theorem 3.1.

Theorem 3.1 (P. Jordan, J. von Neumann, 1935): a normed space $X$ is Euclidean if and only if for any two vectors $x, y \in X$ the parallelogram equality $\left\| x-y\right\|^{2}+\left\| x+y\right\|^{2} = 2\left\| x\right\|^{2}+2\left\| y\right\|^{2}$ holds.

Prove that if in a normed space $X$ for arbitrary two vectors $x, y$ the "parallelogram inequality" $\left\| x-y\right\|^{2}+\left\| x+y\right\|^{2} \leqslant 2\left\| x\right\|^{2}+2\left\| y\right\|^{2}$ holds, then $X$ is Euclidean.

Prove that the following spaces are not Hilbert spaces:

Prove that in any Banach space $X$, for any vectors $x, y \in X$ not both equal to zero, $\frac{1}{2} \leqslant \frac{\left\| x+y\right\|^{2}+\left\| x-y\right\|^{2}}{2\left\| x\right\|^{2}+2\left\| y\right\|^{2}} \leqslant 2$ holds. Show that in both spaces $l_{\infty }(2)$ and $C[0,1]$ both extreme values are attained. ¹

Prove that the space $L_{2}(M, \Sigma , \mu )$ (see the list of spaces) is a Hilbert space.

Let $\Omega = \left\{ r_{n}\right\}_{1}^{\infty }$ be a sequence of positive numbers. Consider the set of sequences $(x_{1}, x_{2}, \ldots )$, where $x_{n} \in \mathbb {C}$, for which the series $\sum_{n = 1}^{\infty }\left(r_{n}\left|x_{n}\right|\right)^{2}$ converges. Prove that this is a linear space. Equip this set with the inner product (verify the axioms of an inner product)

and denote the resulting space $l_{2, \Omega }$. Prove that this space is $L_{2}(M, \Sigma , \mu )$, where $M = \mathbb {N}, \Sigma = 2^{\mathbb {N}}$, and $\mu (A) = \sum_{n \in A} r_{n}$. It follows that $l_{2, \Omega }$ is a Hilbert space. ¹

Prove that the Sobolev spaces $W_{2}^{n}[0,1], n \in \mathbb {N}$, are Hilbert spaces.

Prove that the Bergman space $\mathcal{A} \mathcal{L}_{2}(\left|z\right| < 1)$ is a Hilbert space.

Which of the following spaces are separable:

Consider the linear space $H$ of arbitrary functions of a real variable that are nonzero at no more than a countable number of points (the set of points is different for each function). Introduce an inner product on this space by the rule $(x, y) = \sum_{t: x(t) \cdot y(t) \neq 0} x(t) \overline{y(t)}$. Prove that $H$ is a nonseparable Hilbert space.

Prove that every Hilbert space is strictly convex, or, equivalently, strictly normed (see Definition 2.4).

Definition 2.4: A normed space $X$ is called strictly normed (or strictly convex) if for $\left\| x\right\| = \left\| y\right\| = 1$ the equality $\left\| x+y\right\| = \left\| x\right\| +\left\| y\right\|$ holds if and only if $x = y$ (in other words, the unit sphere of $X$ contains no line segments).

Prove that every Hilbert space is uniformly convex (see Definition 2.11).

Definition 2.11: A normed space $X$ is called uniformly convex if for every $\varepsilon > 0$ there exists $\delta > 0$ such that if $\left\| x\right\| = \left\| y\right\| = 1$ and $\left\| x+y\right\| > 2-\delta$, then $\left\| x-y\right\| < \varepsilon$.

Prove that in a complex Hilbert space $H$, for any vectors $x$ and $y$, the equalities hold

(compare with problem 3.6).