Math Problems Vault

Problem 24

(a)(b)

Let $M$ be a closed subspace of a Hilbert space $H$ . Prove that:

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(a)

$\left(M^{\perp }\right)^{\perp } = M$ .

(b)

If $\operatorname {codim} M: = \operatorname {dim} H / M = 1$ then $\operatorname {dim} M^{\perp } = 1$ .

Problem 25

Let $T: H_{1} \rightarrow H_{2}$ be an isometry of two Hilbert spaces $H_{1}$ and $H_{2}$ , i.e., $\left\| T x\right\| = \left\| x\right\|$ for every $x \in H_{1}$ . Prove that

\langle T x, T y\rangle = \langle x, y\rangle \quad \text{ for every } x, y \in H_{1}.

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Problem 26

Let $C$ be a closed convex set in a Hilbert space $H$ . Show that $C$ contains a unique element of minimal norm.

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Problem 27

Let $1 \leq p < \infty$ . Prove that $\ell^{p}$ is a Hilbert space if and only if $p = 2$ .

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Problem 28

(a)(b)

Consider the Hilbert space $H = L^{2}[-1,1]$ equipped with the usual scalar product:

\langle x, y\rangle = \int _{-1}^{1} \overline{x(t)} y(t) d t, \quad x, y \in H

Let $M = \left\{ x \in H: \int_{-1}^{1} x(t) d t = 0\right\}$ .

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(a)

Show that $M$ is closed in $H$ . Find $M^{\perp }$ .

(b)

Calculate the distance from $y$ to $M$ for $y(t) = t^{2}$ .

Problem 29

(a)(b)

Consider the Hilbert space $H = L^{2}[-1,1]$ equipped with the usual scalar product:

\langle f, g\rangle = \int _{-1}^{1} \overline{f(t)} g(t) d t, \quad f, g \in H.

Let $E = \left\{ x \in H: f(-t) = f(t), \quad t \in [-1,1]\right\}$ .

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(a)

Show that $E$ is closed in $H$ . Find $E^{\perp }$ .

(b)

Calculate the distance from $h$ to $E$ for $h(t) = e^{t}$ .

Problem 30

Let $H$ be a Hilbert space and $M$ be a closed subspace of $H$ . Denoting by $P$ : $H \rightarrow M$ the orthogonal projection of $H$ onto $M$ , prove that, for any $x, y \in H$ ,

\langle P x, y\rangle = \langle x, P y\rangle .

(This is telling us that $P$ is self-adjoint).

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Problem 31

Let $H$ be a Hilbert space and $A \subset H$ a closed convex non-empty set. Prove that $P_{A}: H \rightarrow H$ is non-expansive, i.e.,

\left\| P_{A}(x)-P_{A}(y)\right\| \leq \left\| x-y\right\| , \quad \forall x, y \in H.

( $P_{A}$ is the orthogonal projection on $A$ ).

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Problem 32

(a)(b)

Let $X$ be a Hilbert space, and $G_{1} \subset G_{2} \subset \ldots \subset G_{n} \subset \ldots$ be a sequence of closed linear subspaces of $X$ . Let

G = \overline{\operatorname {Span}\left(\bigcup _{n \in \mathbb {N}} G_{n}\right)}.

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(a)

Prove that $d(x, G) = \lim_{n \rightarrow \infty } d\left(x, G_{n}\right), \forall x \in X$ .

(b)

Prove that $P_{G}(x) = \lim_{n \rightarrow \infty } P_{G_{n}}(x), \forall x \in X$ . Note: $P_{G}$ is the orthogonal projection of $X$ on $G$ .

Problem 33

(a)(b)

Let $H$ be a Hilbert space.

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(a)

Prove that for any two subspaces $M, N$ of $H$ we have

(M+N)^{\perp } = M^{\perp } \cap N^{\perp }

(b)

Prove that for any two closed subspaces $E, F$ of $H$ we have

(E \cap F)^{\perp } = \overline{E^{\perp }+F^{\perp }}

Remark.

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Why in question (b) must $E$ and $F$ be closed?

Problem 34

A system $\left\{ x_{i}\right\}_{i \in \mathbb {N}}$ in a normed space $X$ is called a complete system if $\operatorname {Span}\left\{ \sum_{i = 1}^{n} \alpha_{i} x_{i}: \quad \forall n \in \mathbb {N}, \alpha_{i} \in \mathbb {F}\right\}$ is dense on $X$ . If $\left\{ x_{i}\right\}_{i \in \mathbb {N}}$ is a complete system in a Hilbert space $H$ and $x \perp x_{i}$ for every $i$ , show that $x = 0$ .

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Problem 35

Let $H$ be a Hilbert space and $\left\{ \varphi_{i}\right\}_{i = 1}^{\infty }$ an orthonormal system in $H$ . Show that

\left\| \varphi _{n}-\varphi _{m}\right\| = \sqrt{2} \text{ for } m \neq n.

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Problem 36

Let $\left\{ e_{i}\right\}_{i \in \mathbb {N}}$ be an orthonormal set in a Hilbert space H. Prove that

\sum _{i = 1}^{\infty }\left| \left\langle x, e_{i} \right\rangle \left\langle y, e_{i} \right\rangle \right| \leq \left\| x\right\| \left\| y\right\| , \quad \forall x, y \in H

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Problem 37

(a)(b)(c)(d)

Let $H$ be a Hilbert space and $A$ and $B$ be any two subsets of $H$ . Show that

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(a)

$A^{\perp }$ is a closed subspace of $H$ .

(b)

$A \subset \left(A^{\perp }\right)^{\perp }: = A^{\perp \perp }$ .

(c)

$A \subset B \Rightarrow B^{\perp } \subset A^{\perp }$ .

(d)

$A^{\perp } = \bar{A}^{\perp } = \overline{A^{\perp }}$ .

Problem 38

(a)(b)

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(a)

Show that $M: = \left\{ x = \left(x_{n}\right) \in \ell^{2}: x_{2 n} = 0, \forall n \in \mathbb {N}\right\}$ is a closed subspace of the Hilbert space $\ell^{2}$ .

(b)

Find $M^{\perp }$ .

Problem 39

(a)(b)

Let $V$ be a subspace of a Hilbert space.

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(a)

Show that $\bar{V}^{\perp } = V^{\perp }$ .

(b)

Show that $V$ is dense in $H$ if and only if $V^{\perp } = \left\{ 0\right\}$ .

Problem 40

Prove that in any finite dimensional vector space, strong convergence and weak convergence are equivalent.

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Problem 41

Show that if the sequence $(x_{n})$ in a normed space $X$ is weakly convergent to $x_{0} \in X$ , then

\liminf _{n \rightarrow \infty }\left\| x_{n}\right\| \geq \left\| x_{0}\right\| .

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Problem 42

Let $X$ and $Y$ be normed spaces, $T \in \mathcal{B}(X, Y)$ and $\left(x_{n}\right)$ a sequence in $X$ . Show that if $x_{n} \xrightarrow {w} x$ , then $T x_{n} \xrightarrow {w} T x$

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Problem 43

Let $H$ be a Hilbert space and $\left(x_{n}\right)$ be a sequence in $H$ . Suppose $x_{n} \xrightarrow {w} x$ . Show that

\lim _{n \rightarrow \infty }\left\| x_{n}-x\right\| = 0 \Longleftrightarrow \left\| x\right\| \geq \limsup _{n \rightarrow \infty }\left\| x_{n}\right\| \tag {1}

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