Math Problems Vault

Problem 462

(a)(b)

?

(a)

Prove that the correspondence $L\to \widetilde L$ constructed in 4.1 of Chapter III of the Theory part defines a covariant functor from the category of pre-Hilbert spaces to the category of Hilbert spaces.

(b)

Show that $\widetilde L$ is a universal object in a suitable category.

Here a pre-Hilbert space may have a positive semidefinite scalar product. Set $L_0=\left\{ x\in L: \left\langle x, x \right\rangle =0\right\}$ , give $L/L_0$ the induced scalar product, and let $\widetilde L$ be its completion in the norm $\left\| x\right\| = \left\langle x, x \right\rangle^{1/2}$ . The scalar product extends continuously to $\widetilde L$ .

Problem 463

Prove that the system of functions $e_n(x)=e^{2\pi inx}$ , $n\in \mathbb {Z}$ , is a Hilbert basis in $L^{2}\left({(0,1)}\right)$ .

?

Problem 464

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

Apply the orthogonalization process to the sequence of monomials $1,x,x^2,\ldots$ in the following Hilbert spaces:

?

(a)

$L^{2}\left({[-1,1],dx}\right)$ ;

(b)

$L^{2}\left({[-1,1],\frac{dx}{\sqrt{1-x^2}}}\right)$ ;

(c)

$L^{2}[{[0,\infty ),e^{-x}\, dx}]$ ;

(d)

$L^{2}\left({(-\infty ,\infty ),e^{-x^2}\, dx}\right)$ .

Problem 465

(a)(b)

Apply the orthogonalization process to the sequence of monomials $1,z,z^2,\ldots$ in the pre-Hilbert spaces of polynomials with the following scalar products:

?

(a)

$\displaystyle \left\langle P, Q \right\rangle =\iint_{\left|z\right|\le R}P(z)\overline{Q(z)}\, dx\, dy$ ;

(b)

$\displaystyle \left\langle P, Q \right\rangle =\iint_{\mathbb {C}}P(z)\overline{Q(z)}e^{-\left|z\right|^2}\, dx\, dy$ .

Problem 466

Express the linear functional $F_x(f)=f(x)$ in the spaces of Problem 465 in the form of a scalar product.

?

Problem 467

[NO TASK]

Problem 468

(a)(b)

Prove that the orthogonal complement of the system of functions $e_n(x)=e^{2\pi inx}$ , $n\in \mathbb {Z}$ , in the space $L^{2}\left({[a,b],dx}\right)$

?

(a)

is equal to zero for $\left|b-a\right|\le 1$ ;

(b)

is different from zero for $\left|b-a\right|>1$ .

Problem 469

(a)(b)

Let $B_0$ be the space of all trigonometric polynomials (i.e., finite linear combinations of the functions $e^{i\lambda x}$ , $\lambda \in \mathbb {R}$ ). Define in $B_0$ the scalar product

\left\langle f, g \right\rangle =\lim _{A\to +\infty }\frac1{2A}\int _{-A}^A f(x)\overline{g(x)}\, dx.

?

(a)

Prove that the corresponding Hilbert space $B$ is nonseparable and has the continuum Hilbert basis $\left\{ e^{i\lambda x}\right\}_{\lambda \in \mathbb {R}}$ .

(b)

Prove that $B$ contains the space of almost periodic continuous functions, i.e., the closure of $B_0$ in the uniform norm $\left\| f\right\| =\sup_{x\in \mathbb {R}}\left|f(x)\right|$ , but does not coincide with it.

Problem 470

Let $\mu$ be the measure on the line defined by the formula $\mu (A)=\operatorname {card}A$ (the number of points in $A$ ). Prove that $L^{2}\left({\mathbb {R},\mu }\right)$ is nonseparable and isomorphic to the space of Problem 469.

?

Problem 471

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

Prove that $L^{2}\left({[a,b]}\right)$ contains complete orthonormal systems consisting of:

?

(a)

polynomials;

(b)

step functions;

(c)

trigonometric polynomials;

(d)

functions lying in any given dense subspace.

Problem 472

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

Find the orthogonal complements in $L^{2}\left({(1,0)}\right)$ of the following sets:

?

(a)

the polynomials in $x$ ;

(b)

the polynomials in $x^2$ ;

(c)

the polynomials with zero free term;

(d)

the polynomials with sum of coefficients equal to zero.

Problem 473

(a)(b)

Find the orthogonal complements of the following spaces in the pre-Hilbert space $C[-1,1]$ of all continuous functions on $[-1,1]$ with the scalar product $\left\langle f, g \right\rangle =\int_{-1}^1 f(x)\overline{g(x)}\, dx$ :

?

(a)

the functions equal to zero for $x\le 0$ ;

(b)

the functions equal to zero at the point $x=0$ .

Is the theorem on orthogonal complements true in these cases? The theorem on orthogonal complements states that if $H_1$ is a closed subspace of a Hilbert space $H$ and $H_2=H_1^\perp$ , then $H=H_1\oplus H_2$ .

Problem 474

(a)(b)

Let $e_t$ be the characteristic function of the closed interval $[0,t]$ , $t\ge 0$ . Find the angles between two chords of the curve $e_t$ in $L^{2}\left({(0,\infty )}\right)$ :

?

(a)

if the chords have a common endpoint and opposite directions with respect to $t$ ;

(b)

if the chords have a common endpoint and the same direction with respect to $t$ .

Problem 475

(a)(b)

?

(a)

Prove that in every pre-Hilbert space the parallelogram law is satisfied:

\left\| x-y\right\| ^2+\left\| x+y\right\| ^2=2\left\| x\right\| ^2+2\left\| y\right\| ^2.

(b)

Prove that in every normed linear space over $\mathbb {R}$ or $\mathbb {C}$ in which this law is satisfied it is possible to introduce a scalar product such that $\left\| x\right\|^2= \left\langle x, x \right\rangle$ .

Problem 476

(a)(b)

Prove that in a complex Hilbert space the following equalities hold:

?

(a)

$\displaystyle \left\langle x, y \right\rangle =\frac1N\sum_{k=1}^N\left\| x+e^{2\pi ik/N}y\right\|^2e^{2\pi ik/N}$ for $N\ge 3$ ;

(b)

$\displaystyle \left\langle x, y \right\rangle =\frac1{2\pi }\int_0^{2\pi }\left\| x+e^{i\theta }y\right\|^2e^{i\theta }\, d\theta$ .

Problem 477

Suppose that the family $\left\{ x_n\right\}$ of vectors in a Hilbert space $H$ has the properties that $\left\| x_n\right\| =1$ , $\left\langle x_n, x_m \right\rangle =c$ for $m\ne n$ . Prove that the sequence $\left\{ x_n\right\}$ has a weak limit.

?

Problem 478

Prove that the following conditions are equivalent for an orthogonal system $\left\{ x_n\right\}$ of vectors:

$\sum_nx_n$ is strongly convergent;
$\sum_nx_n$ is weakly convergent;
$\sum_n\left\| x_n\right\|^2$ converges.

?

Problem 479

Let $S$ be any subset of a Hilbert space $H$ . Prove that $(S^\perp )^\perp$ coincides with the closure of the linear span of $S$ .

?

Problem 480

Let $L$ be a subspace of a Hilbert space $H$ , and $f_0$ a continuous linear functional on $L$ . Prove that $f_0$ has a unique extension to $H$ with the same norm.

?