Hilbert Spaces

Prove that a scalar product in a Hilbert space is a continuous function in each of the arguments.

Let $L$ and $M$ be linear sets in a Hilbert space $H$, and $L \perp M$.

Find the orthogonal complement in $L_{2}(\left[-1,1\right])$ of the set of all even functions.

Let $L$ be a total set in a Hilbert space $H$. Prove that $L^{\perp } = \left\{ 0\right\}$.

Find the orthogonal complement of the set $M = \left\{ e^{k t} \mid k \geq 10\right\}$ in $L_{2}(\left[0,1\right])$.

Let $\alpha = \left(\alpha_{1}, \alpha_{2}, \ldots \right)$, where $\alpha_{k} > 0, k \geq 1$. Consider the set $l_{2, \alpha }$ of all numerical sequences $x = \left(x_{1}, x_{2}, \ldots \right)$ for which $\sum_{k = 1}^{\infty } \alpha_{k}\left|x_{k}\right|^{2} < \infty$. Verify that $l_{2, \alpha }$ is a Hilbert space with the scalar product $(x, y) = \sum_{k = 1}^{\infty } \alpha_{k} x_{k} \overline{y_{k}}, x, y \in l_{2, \alpha }$.

Construct an orthonormal basis in the space $l_{2, \alpha }$ in the following cases:

Determine for which $\alpha = \left(\alpha_{1}, \alpha_{2}, \ldots \right)$, where $\alpha_{k} > 0, k \geq 1$, the inclusion $l_{2, \alpha } \subset l_{2}$ holds.

Prove that in the Banach spaces $c_{0}, l_{p}, L_{p}(\left[a, b\right]), p \neq 2$, the norm is not generated by a scalar product.

Prove that for arbitrary $x, y, z$ in a Hilbert space Apollonius identity holds:

Prove that for arbitrary $x, y, z, u$ in a Hilbert space Ptolemy's inequality holds:

Prove that in a Hilbert space over a field $\mathbb {K}$ the elements $x$ and $y$ are orthogonal if and only if

Prove that the polarization identity holds in a complex Hilbert space:

What does it look like in a real Hilbert space?

Let $X$ be a complex NLS, and let the function $S: X \times X \rightarrow \mathbb {C}$ has the following properties:

Let $X$ be a complex NLS in which the norm satisfies the parallelogram law $\left\| x+y\right\|^{2}+\left\| x-y\right\|^{2} = 2\left(\left\| x\right\|^{2}+\left\| y\right\|^{2}\right)$ for all $x, y \in X$. Prove that the formula

$x, y \in X$, defines a scalar product in the space $X$, and $\left\| \cdot \right\|$ is the norm generated by this scalar product.

Let $p$ be a Lebesgue-measurable function on $(a, b)$ such that $p(t) > 0$ for almost all $t \in (a, b)$, and $L_{2, p}(\left[a, b\right]) = \left\{ x \left\lvert \, p^{\frac{1}{2}} x \in L_{2}(\left[a, b\right])\right.\right\}$. Prove that $L_{2, p}(\left[a, b\right])$ is a Hilbert space with scalar product $(x, y) = \int_{a}^{b} x(t) \overline{y(t)} p(t) d t$. For which $p$ the inclusion $L_{2, p}(\left[a, b\right]) \subset L_{2}(\left[a, b\right])$ holds?

Let $\left\{ x_{k}: k \geq 1\right\}$ be an orthogonal system in a Hilbert space $H$. Prove that the series $\sum_{k = 1}^{\infty } x_{k}$ converges in $H$ if and only if $\sum_{k = 1}^{\infty }\left\| x_{k}\right\|^{2} < \infty$.

Prove that a finite system $\left\{ x_{1}, \ldots , x_{n}\right\}$ of elements of a Hilbert space is linearly independent if and only if its Gram determinant

is nonzero.

Let $H$ be a Hilbert space, $x, y \in H,\left\| x\right\| = \left\| y\right\| = 1, a, b \in \mathbb {R}$. Prove that $\left\| a x+b y\right\| = \left\| b x+a y\right\|$.

Let $H$ be a Hilbert space, $\left\{ x_{n}\right\} \subset H,\left\{ y_{n}\right\} \subset H$, and $\left\| x_{n}\right\| = \left\| y_{n}\right\| = 1, n \geq$ 1. Prove that

Prove that an element $x$ of a Hilbert space $H$ is orthogonal to a subspace $L$ if and only if $\left\| x\right\| \leq \left\| x-y\right\|$ for all $y \in L$.

Prove that for an arbitrary set $M \subset H$ the orthogonal complement $M^{\perp }$ is a subspace.

Let $M \subset N \subset H$. Prove that $M^{\perp } \supset N^{\perp }$.

Prove that

Let $H$ be a Hilbert space. Prove that

Let $L = \left\{ x \in L_{2}(\mathbb {R}) \mid x(t) = 0, t \geq 0(\bmod m)\right\}$. Prove that $L$ is a subspace of $L_{2}(\mathbb {R})$ and find the orthogonal complement of $L$.

Let $L \subset H$. Prove that $\overline{\operatorname {span}(L)} = H$ if and only if $L^{\perp } = \left\{ 0\right\}$.

Find in $L_{2}(\left[0,1\right])$ the orthogonal complement of the set

Find in $L_{2}(\left[-1,1\right])$ the orthogonal complement of the set

Find in $L_{2}(\left[-\pi , \pi \right])$ the orthogonal complement of the set

Find in $l_{2}$ the orthogonal complement of the set

Prove that the function

defines a scalar product in the linear space $C^{1}(\left[a, b\right])$. The completion of the space $C^{1}(\left[a, b\right])$ in the norm generated by this scalar product is a Hilbert space called Sobolev space $W_{2}^{1}(\left[a, b\right])$.

Find the orthogonal complement of the set $\left\{ e^{i k t} \mid k \in \mathbb {Z}\right\}$ in the space $W_{2}^{1}(\left[-\pi , \pi \right])$.

Consider the space $H$ that consists of the functions $x: \mathbb {R} \rightarrow \mathbb {R}$, for which $\left\{ x \in \mathbb {R} \mid x(t) \neq 0\right\}$ is a countable set and $\sum_{t \in \mathbb {R}}\left|x(t)\right|^{2} < +\infty$. Prove that $H$ is a nonseparable Hilbert space with the scalar product $(x, y) = \sum_{t \in \mathbb {R}} x(t) y(t), x, y \in H$.

Prove that the system of functions $\left\{ e^{-i n t} \mid n \in \mathbb {Z}\right\}$ is complete in $L_{2}(\left[a, b\right])$ if and only if $b-a \leq 2 \pi$.

Let $L$ be a linear span of the set $\left\{ e^{i \lambda t} \mid \lambda \in \mathbb {R}\right\}$ equipped with the scalar product $(x, y) = \lim_{T \rightarrow \infty } \frac{1}{2 T} \int_{-T}^{T} x(t) \overline{y(t)} d t, x, y \in L$. Prove that

Prove that the system of Haar functions

where $x_{k n}(t) = \begin{cases} 2^{\frac{n}{2}}, & t \in \left[\frac{k-1}{2^{n}}, \frac{k-1 / 2}{2^{n}}\right), \\ -2^{\frac{n}{2}}, & t \in \left[\frac{k-1 / 2}{2^{n}}, \frac{k}{2^{n}}\right), \\ 0, & t \in [0,1] \backslash \left[\frac{k-1}{2^{n}}, \frac{k}{2^{n}}\right), \end{cases} \quad x_{0}(t) = 1, t \in [0,1]$, is an orthonormal basis in the space $L_{2}(\left[0,1\right])$.

Verify that the following systems are orthogonal in $H$ :

Prove that in problem 2.38

In the space $L_{2}(\mathbb {R})$, consider the set

where $C_{0}(\mathbb {R})$ is the set of continuous functions on $\mathbb {R}$ which vanish outside some finite interval, see problem $1.54^{\circ }$. Prove that $M$ is dense in $L_{2}(\mathbb {R})$.

In the space $l_{2}$, find the orthogonal complement of the set

Let $M \subset H$, where $H$ is a Hilbert space. Is the formula $H = M \oplus M^{\perp }$ always true?

Let $\left\{ e_{n}: n \geq 1\right\}$ be a standard basis in $l_{2}, x_{1} = \sum_{k = 1}^{\infty } \frac{e_{k}}{k}$ and $H = \operatorname {span}\left(\left\{ x_{1}, e_{k} : k \geq 2\right\} \right) \subset l_{2}$. Prove that in the pre-Hilbert space $H$ the set $M = \left\{ e_{n}: n \geq 2\right\}$ is a complete orthonormal system, but $\overline{\operatorname {span}(M)} \neq H$.

Prove that a separable space equipped with a scalar product is complete if and only if every complete orthonormal system in this space is a basis.

Prove that in a Hilbert space there exists a dense linear subset, which does not coincide with the entire space, if and only if the space is infinite-dimensional.

Let $\left\{ e_{n}: n \geq 1\right\}$ be linearly independent system of elements of a Hilbert space $H$ such that $\sum_{n = 1}^{\infty } c_{n} e_{n} = 0$ for some sequence $\left\{ c_{n}: n \geq 1\right\} \subset \mathbb {K}$. Is it true that $c_{n} = 0, n \geq 1$?

Let $H$ be a Hilbert space over a field $\mathbb {K},\left\{ e_{n}: n \geq 1\right\} \subset H$ be an orthonormal system and $x = \sum_{n = 1}^{\infty } x_{n} e_{n}$, where $x_{n} \in \mathbb {K}, n \geq 1$. Prove that $x_{n} = \left(x, e_{n}\right), n \geq 1$.

Prove that the system $\left\{ t^{n}, t \in [0,1]: n \geq 0\right\}$ is linearly independent, and its closed linear span is equal to $L_{2}(\left[0,1\right])$, but not for all $x \in L_{2}(\left[0,1\right])$ there exist coefficients $\left\{ c_{n}: n \geq 0\right\}$ such that $x(t) = \sum_{n = 0}^{\infty } c_{n} t^{n}$ (the convergence here is in the sense of convergence in $L_{2}(\left[0,1\right])$.) Does this contradict to the statement of Theorem 2.6? Why?

Theorem 2.6: Let $\left\{ e_{n}: n \geq 1\right\}$ be an orthonormal sequence in a Hilbert space $H$. Then the following conditions are equivalent:

Let $D = \bar{B}(0,1) \subset \mathbb {C}$ be the closed unit circle in the complex plane, $A_{2}(D)$ be the set of functions $x \in L_{2}(D)$, for which $x(z) = \sum_{n = 0}^{\infty } a_{n} z^{n}$ for $\left|z\right| < 1$. Prove that $A_{2}(D)$ is the completion of the set $P(D)$ of all polynomials defined on $D$, in the norm of $L_{2}(D)$.

Apply the orthogonalization process to the sequence $\left\{ z^{n}: n \geq 0\right\} , z = t+i s$, with respect to the following scalar product on the set of all polynomials $P(\mathbb {C})$ :

Describe the completion of the set $P(\mathbb {C})$.

Let $\left\{ \varphi_{n}: n \geq 1\right\}$ and $\left\{ \psi_{n}: n \geq 1\right\}$ be complete orthonormal systems in $L_{2}\left(X_{1}, \mathcal{F}_{1}, \mu_{1}\right)$ and $L_{2}\left(X_{2}, \mathcal{F}_{2}, \mu_{2}\right)$, respectively, and $\mu_{i}$ be $\sigma$-finite measures on $\mathcal{F}_{i}, i = 1,2$. Prove that $e_{n, m}(x, y) = \varphi_{n}(x) \psi_{m}(y),(x, y) \in X = X_{1} \times X_{2}, n, m \geq$ 1, is a complete orthonormal system in $L_{2}(X, \mathcal{F}, \mu )$, where $\mu = \mu_{1} \times \mu_{2}, \mathcal{F} = \sigma \left(\mathcal{F}_{1} \times \mathcal{F}_{2}\right)$.

Prove that a Hilbert space is uniformly convex (see problem 1.98).

Let $M$ be a nonempty closed convex set in a Hilbert space $H$. Prove that in $M$ there exists a unique element with the smallest norm. Show that the convexity condition is essential.

Let $H$ be a Hilbert space, $x_{0} \in H, r > 0$. For $x \in H \backslash S\left(x_{0}, r\right)$ find $y \in S\left(x_{0}, r\right)$ such that $\rho \left(x, S\left(x_{0}, r\right)\right) = \left\| x-y\right\|$.

Let $x \in H$ and $M \subset H$ be a closed convex set in a real Hilbert space $H$. Prove that $y \in M$ satisfies the condition $\rho (x, M) = \left\| x-y\right\|$ if and only if $(x-y, y-z) \geq 0$ for all $z \in M$.

Let $L$ be a one-dimensional subspace of a Hilbert space $H, a \in L, a \neq 0$. Prove that $\rho \left(x, L^{\perp }\right) = \frac{\left|(x, a)\right|}{\left\| a\right\| }$ for all $x \in H$.

For the set $M_{n} = \left\{ x \in l_{2} \mid \sum_{k = 1}^{n} x_{k} = 0\right\} , n \in \mathbb {N}$, find $M_{n}^{\perp }, \rho \left(x_{0}, M_{n}\right)$ and $\lim_{n \rightarrow \infty } \rho \left(x_{0}, M_{n}\right)$, where $x_{0} = (1,0, \ldots )$.

Let $M, N$ be subspaces of a Hilbert space $H$ and for each $x \in H$ there exist unique $x_{1} \in M$ and $x_{2} \in N$ such that $x = x_{1}+x_{2}$. Is it true that $N = M^{\perp }$?

Let

Prove that $M$ and $N$ are subspaces of $l_{2}$ and the set $M+N$ is dense in $l_{2}$, but $M+N \neq l_{2}$, i.e. $M+N$ is not a subspace of $l_{2}$.

Let $\left\{ e_{n}: n \geq 1\right\}$ be an orthonormal basis in $H, u_{n} = \cos \frac{1}{n} \cdot e_{2 n}+\sin \frac{1}{n} \cdot e_{2 n+1}$. Prove that $\left\{ u_{n}: n \geq 1\right\}$ is an orthonormal system. Put $M = \overline{\operatorname {span}\left(\left\{ e_{2 n}: n \geq 1\right\} \right)}$ and $N = \operatorname {span}\left(\left\{ u_{n}: n \geq 1\right\} \right)$. Prove that the linear set $M+N$ is not closed.

Let $M$ and $N$ be subspaces of a Hilbert space $H$. Prove that $M+N$ is a subspace of $H$, if the following condition holds:

in particular if $M \perp N$.

Let $\left\{ x_{n}: n \geq 1\right\} ,\left\{ y_{n}: n \geq 1\right\}$ be orthonormal systems in $L_{2}(\left[a, b\right])$ and $L_{2}(\left[b, c\right])$ respectively, $\lambda > 0, \mu > 0, \lambda^{2}+\mu^{2} = 1$. Put

Prove that $\left\{ z_{n}: n \geq 1\right\}$ is an orthonormal system in $L_{2}(\left[a, c\right])$. Is $\left\{ z_{n}: n \geq 1\right\}$ necessarily an orthonormal basis in $L_{2}(\left[a, c\right])$, if $\left\{ x_{n}: n \geq 1\right\}$ and $\left\{ y_{n}: n \geq 1\right\}$ are orthonormal bases in $L_{2}(\left[a, b\right])$ and $L_{2}(\left[b, c\right])$, respectively?

Let $a < b < c$ and $\left\{ x_{n}: n \geq 1\right\}$ be an arbitrary countable system of functions from $L_{2}(\left[a, b\right])$. Prove that the functions $x_{n}$ can be extended to the segment $\left[b, c\right]$ to obtain an orthogonal system in $L_{2}(\left[a, c\right])$.

Show that the functions $x_{n, k}(t) = \sqrt{2} \chi_{\left[k-1, k\right]}(t) \sin \pi n t$ for $n \in \mathbb {N}, k \in \mathbb {N}$ form an orthonormal basis in $L_{2}([0,+\infty ))$, and for $n \in \mathbb {N}, k \in \mathbb {Z}$ they form an orthonormal basis in $L_{2}(\mathbb {R})$.