Math Problems Vault

Problem 10.1

Determine whether the following sets are precompact (compact) in the respective spaces:

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(1)

$M = \left\{ e_{n}: n \geq 1\right\}$ in $l_{p}, 1 \leq p < +\infty ;$

(2)

$M = \left\{ x_{n}(t) = e^{-n t}, t \in [0,1]: n \geq 1\right\}$ in $C(\left[0,1\right])$ , in $L_{p}(\left[0,1\right]), 1 \leq p < +\infty ;$

(3)

$M = \left\{ x_{\alpha }(t) = \cos \alpha t, t \in [0, \pi ] \mid \alpha \in [0,2]\right\}$ in $C(\left[0, \pi \right])$ , in $L_{p}(\left[0, \pi \right]), 1 \leq p < +\infty ;$

(4)

$M = \left\{ x_{n}(t) = \cos n t, t \in [0, \pi ]: n \geq 1\right\}$ in $C(\left[0, \pi \right])$ , in $L_{p}(\left[0, \pi \right]), 1 \leq p < +\infty ;$

(5)

$M$ is a bounded set in $C(\left[0,1\right])$ , which consists of polynomials of degree at most $n$ , where $n$ is fixed;

(6)

$M = \left\{ x \in C^{1}(\left[0,1\right]) \mid \int_{0}^{1}\left|x(t)\right| d t \leq 1, \left|x^{\prime }(t)\right| \leq 3, t \in [0,1]\right\}$ in $C(\left[0,1\right])$ .

Problem 10.2

Let $H = L_{2}(\left[a, b\right]),\left\{ p_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{ q_{k} \mid 1 \leq k \leq n\right\}$ be finite sets of functions from $L_{2}(\left[a, b\right])$ . Prove that the operator defined by the formula

(A x)(t) = \int _{a}^{b}\left(\sum _{k = 1}^{n} p_{k}(t) q_{k}(s)\right) x(s) d s

is compact in $H$ .

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Problem 10.3

Let $\left\{ a_{n}: n \geq 1\right\}$ be a bounded sequence of complex numbers. Consider the operator $A: l_{2} \rightarrow l_{2}, A x = \left(a_{1} x_{1}, a_{2} x_{2}, \ldots \right)$ . Prove that $A$ is compact if and only if $\lim_{n \rightarrow \infty } a_{n} = 0$ .

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Problem 10.4

Let $K \in C\left(\left[a, b\right]^{2}\right)$ and $(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s, t \in [a, b], x \in C(\left[a, b\right])$ . Prove that $A \in S_{\infty }(C(\left[a, b\right]))$ .

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