Math Problems Vault

Problem 2.89

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

Prove that the set of finitely supported sequences:

?

(a)

is dense in the space $l_{p}, p \in [1, \infty )$ ;

(b)

is dense in the space $c_{0}$ ;

(c)

is not dense in the space $c$ ;

(d)

is not dense in the space $l_{\infty }$ .

Problem 2.90

Prove that the set of polynomials:

?

(a)

is dense in the space $C[0,1] ;$

(b)

is dense in the space $C^{n}[0,1]$ ;

(c)

is dense in the space $C_{p}[0,1]$ , where $p \in [1, \infty ) ;$

(d)

is dense in the space $C_{p}^{n}[0,1]$ , where $n = 1,2, \ldots$ , and $p \in [1, \infty )$ ;

(e)

is not dense in the space $L_{\infty }[0,1]$ ;

(f)

is not dense in the space $B V[0,1]$ .

Problem 2.91

Let $\mu$ be an arbitrary finite measure on some $\sigma$ -algebra $\Sigma$ of subsets of the segment $[0,1]$ . Prove that the set of polynomials is dense in the spaces $L_{p}([0,1], \Sigma , \mu ), p \in [1, \infty )$ .

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Problem 2.92

Prove that the set of piecewise-linear functions:

?

(a)

is dense in the space $C[0,1]$ ;

(b)

is dense in the space $L_{p}[0,1], p \in [1, \infty )$ ;

(c)

is dense in the space $W_{p}^{1}[0,1], p \in [1, \infty )$ ;

(d)

is not dense in the space $L_{\infty }[0,1]$ ;

(e)

is not dense in the space $B V[0,1]$ .

Problem 2.93

In the space $C[0,1]$ , for an arbitrary natural number $n$ , consider the set of functions $F_{n} = \left\{ x \in C[0,1]: \exists t_{0} \in [0,1] \forall t \in [0,1]\left|x(t)-x\left(t_{0}\right)\right| \leqslant n\left|t-t_{0}\right|\right\}$ . Prove that $F_{n}$ is nowhere dense.

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Problem 2.94

Let the normed spaces $X_{1}$ and $X_{2}$ be isomorphic. Prove that they are either both separable, or both not separable.

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Problem 2.95

Let $(X,\left\| \cdot \right\| )$ be a separable normed space, and $X_{0}$ its linear subspace. Prove that the space $(X_{0},\left\| \cdot \right\| )$ is separable.

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Problem 2.96

Prove that if a normed space contains an uncountable system of pairwise disjoint unit balls, then the space is not separable.

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Problem 2.97

Is the set $M = \left\{ x \in l_{1}: x_{n} \in \mathbb {Q} \text{ for all } n \in \mathbb {N}\right\}$ countable?

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Problem 2.98

Determine which of the following spaces are separable:

?

(a)

$l_{p}(n), p \in [1, \infty ], n \in \mathbb {N}$ ;

(b)

$l_{p}, p \in [1, \infty )$ ;

(c)

$c_{0}$ and $c$ ;

(d)

$l_{\infty }$ ;

(e)

$C[0,1]$ ;

(f)

$C^{n}[0,1], n \in \mathbb {N}$ ;

(g)

$\mathcal{P}_{n}[0,1]$ ;

(h)

$C_{p}[0,1], p \in [1, \infty )$ ;

(i)

$W_{p}^{n}[0,1], n \in \mathbb {N}, p \in [1, \infty )$ ;

(j)

$\mathcal{P}[0,1] ;$

(k)

$L_{p}[0,1], p \in [1, \infty )$ ;

(l)

$L_{\infty }[0,1]$ ;

(m)

$B V[0,1]$ and $B V_{0}[0,1]$ ;

(n)

$\mathcal{A C}(D)$ ;

(o)

$L_{p}([0,1], \Sigma , \mu ), p \in [1, \infty )$ , the measure $\mu$ is finite;

(p)

$L_{p}(\mathbb {R}, \Sigma , \mu ), p \in [1, \infty )$ , the measure $\mu$ is $\sigma$ -finite.