Sets in Hilbert spaces

Let $M$ be an arbitrary set in a Hilbert space. Prove that $M^{\perp }$ is a closed linear subspace.

Let $H_{0}$ be a proper closed subspace of a Hilbert space $H$. Prove that a vector $x$ belongs to $H_{0}^{\perp }$ if and only if for every $y \in H_{0}$ the inequality $\left\| x-y\right\| \geqslant \left\| x\right\|$ holds.

Let $H_{0}$ be a proper closed subspace of a Hilbert space $H$, and $x \in H$. Prove that $y \in H_{0}$ is the orthogonal projection of the vector $x$ if and only if $y$ is the element of best approximation for the vector $x$ in the subspace $H_{0}$.

Prove that in a Hilbert space any closed subspace is a subspace of existence (see Definition 2.17).

Definition 2.17: Let $X$ be a normed space and $Y$ a linear subspace of it. The subspace $Y$ is called a subspace of existence if for every element $x \in X$ there exists in $Y$ an element of best approximation, i.e., a vector $y_{0} \in Y$ with $\operatorname {dist}(x, Y) = \left\| x-y_{0}\right\|$.

Let $H = H_{0} \oplus_{\perp } H_{1}$, where $H_{0}$ and $H_{1}$ are closed subspaces, $x \in H$. Denote by $y$ the element of best approximation for $x$ in the subspace $H_{0}$, and by $z$ the element of best approximation for $x$ in the subspace $H_{1}$. Prove that $x = y+z$ and, correspondingly, $\operatorname {dist}\left(x, H_{0}\right) = \left\| z\right\|$, $\operatorname {dist}\left(x, H_{1}\right) = \left\| y\right\| , \operatorname {dist}^{2}\left(x, H_{0}\right) + \operatorname {dist}^{2}\left(x, H_{1}\right) = \left\| x\right\|^{2}$.

Let $H_{0}$ be a closed subspace of a Hilbert space $H$. Prove that $H = H_{0} \oplus_{1} H_{0}^{\perp }$. Is this true in a pre-Hilbert space $H$?

Let $H$ be a Hilbert space, $H_{0}$ a proper closed subspace, $H / H_{0}$ the quotient space. Prove that the mapping $J: H_{0}^{\perp } \rightarrow H / H_{0}, J x = [x]$ is an isometric isomorphism (here $[x]$ is the quotient class from $H / H_{0}$ containing the vector $x$).

Prove that a linear subspace $M$ in a Hilbert space $H$ is everywhere dense in $H$ if and only if $M^{\perp } = \left\{ 0\right\}$.

Prove that in a Hilbert space, for any set $M$, the equality $\left(M^{\perp }\right)^{\perp } = \overline{\operatorname {Lin}(M)}$ (the closure of the linear span of the set $M$) holds.

Let $H$ be a Hilbert space, and let $M$ and $N$ be two mutually orthogonal linear subspaces of it. Prove that their orthogonal direct sum is a topological direct sum. Prove that if they are both closed, then $M \oplus_{\perp } N$ is a closed subspace in $H$.

In a Hilbert space $H$ give examples of closed subspaces $M$ and $N$ such that $H = M \oplus N$, but this topological direct sum is not an orthogonal direct sum.

Let $H$ be a Hilbert space, $\left\{ x_{n}\right\}_{1}^{\infty }$ a sequence of vectors from $H$, and $\left\{ \lambda_{n}\right\}_{1}^{\infty }$ a sequence of complex numbers. Prove that the set $M = \left\{ x \in H:\left(x, x_{k}\right) = \lambda_{k}, k \in \mathbb {N}\right\}$ is either empty or is a closed affine subspace in $H$. ¹

Prove that any closed convex set in a Hilbert space is a Chebyshev set. ¹

Give an example of a sequence of nonempty nested bounded closed sets in the space $l_{2}$ having empty intersection.

Prove that the angle $\widehat{x, M}$ between a subspace $M$ and a vector $x$ in a real Hilbert space $H$ takes values in the interval $\left[0, \frac{\pi }{2}\right]$.

In the space $L_{2}[-1,1]$, for the following sets find the convex hull, the closure, the linear span, the closure of the linear span, and the orthogonal complement: ¹

In the space $W_{2}^{1}[-1,1]$, for the following sets find the convex hull, the closure, the linear span, the closure of the linear span, and the orthogonal complement:

Let $H_{n}, n \in \mathbb {N} \cup \left\{ \infty \right\}$, be the set of vectors $x = \left(x_{1}, x_{2}, \ldots \right)$ of the space $l_{2}$ for which $\sum_{k = 1}^{n} x_{n} = 0$. Find $H_{n}^{\perp }$ and prove that all $H_{n}, n \neq \infty$, are closed subspaces, while $H_{\infty }$ is a non-closed subspace, everywhere dense in $l_{2}$.

In the real spaces $L_{2}[0,1]$ and $W_{2}^{n}[0,1], n \in \mathbb {N}$, find the angle between the vectors $x(t) = 1$ and $y(t) = t$.

In the space $L_{2}[0,1]$ consider the continuous curve $\gamma :[0,1] \rightarrow L_{2}[0,1], \gamma (\tau ) = \chi_{\tau }$, where $\chi_{a}$ is the characteristic function of the segment $[0, a]$. Let $x = \gamma \left(\tau_{2}\right)-\gamma \left(\tau_{1}\right), y = \gamma \left(\tau_{3}\right)-\gamma \left(\tau_{2}\right)$, $0 \leqslant \tau_{1} < \tau_{2} < \tau_{3} \leqslant 1$, — two chords of the curve $\gamma$ with a common endpoint. Prove that $x$ and $y$ are orthogonal. Does such a curve exist in a finite-dimensional Euclidean space?

In the space $L_{2}[0,1]$ find the distance from the vector $x(t) = t^{n}, n \in \mathbb {N}$, to the subspace $H_{0} = \left\{ x \in L_{2}[0,1]: \int_{0}^{1} x(t) d t = 0\right\}$.

Let $H_{0}$ be a finite-dimensional subspace of a Hilbert space $H$, and let $\left\{ h_{k}\right\}_{1}^{n}$ be a linear basis (not necessarily orthogonal) in $H_{0}$. Prove that for any $x \in H$ the element of best approximation in $H_{0}$ has the form

and the distance from $x$ to $H_{0}$ is given by the equality

In particular, if $\left\{ h_{k}\right\}_{1}^{n}$ is an orthonormal basis in $H_{0}$, then the element of best approximation is $y = \sum_{k = 1}^{n}\left(x, h_{k}\right) h_{k}$, and

In the space $l_{2}$ find $\operatorname {dist}\left(e_{1}, H_{n}\right)$, where $e_{1} = (1,0,0, \ldots )$, $H_{n} = \left\{ x \in l_{2}: \sum_{k = 1}^{n} x_{k} = 0\right\} , n \in \mathbb {N}$.

In the space $L_{2}[0,1]$ find the distance from the vector $x(t) = t^{2}$ to the subspace $\mathcal{P}_{1}$ of all linear functions. Find the element of best approximation for $x$ in $\mathcal{P}_{1}$ (compare with problem 2.73).