Примеры с решениями

$X = C(\left[a, b\right]), f(x) = x\left(\frac{a+b}{2}\right)$;

$X = C(\left[a, b\right]), f(x) = x(a)-x(b)$;

$X = C(\left[-1,1\right]), f(x) = \int_{-1}^{1} x(t) \operatorname {sign} t d t$;

$X = C(\left[0,1\right]), f(x) = \int_{0}^{1} t\left|x(t)\right| d t ;$

$X = C(\left[0,1\right])$ с нормой $\left\| x\right\| = \int_{0}^{1}\left|x(t)\right| d t, f(x) = x(1)$;

$X = L_{p}(\left[-1,1\right]), 1 \leq p \leq +\infty , f(x) = \int_{-1}^{1} t^{2} x(t) \operatorname {sign} t d t$;

$X = l_{2}, f(x) = \sum_{n = 1}^{\infty } \frac{1}{2^{n}}\left(x_{n}+x_{n+1}\right)$;

$X = \left\{ x \in l_{2} \mid \sum_{n = 1}^{\infty } x_{n} \cos n \text{ — сходящийся ряд}\right\} , \left\| \cdot \right\|_{X} = \left\| \cdot \right\|_{l_{2}}$,

Ker $f$ — линейное множество, причём если $f \in X^{*}$, то $\operatorname {Ker} f$ является подпространством $X$;

если $z \in X \backslash \operatorname {Ker} f$, то

$X = C(\left[-1,1\right]), f(x) = \int_{-1}^{1} t x(t) d t-x(0)$;

$X = C(\left[a, b\right]), f(x) = x\left(\frac{a+b}{2}\right)$;

$X = C(\left[0,3\right]), f(x) = \int_{0}^{1} x(t) d t+\int_{2}^{3} t x(t) d t$;

$X = l_{2}, f(x) = \sum_{k = 1}^{\infty } \frac{1}{2^{k}}\left(x_{k}+x_{k+1}\right)$;

$X = l_{p}, 1 \leq p \leq +\infty , f(x) = x_{j}+x_{j+1}, j$ фиксировано;

$X = l_{p}, 1 \leq p \leq +\infty , f(x) = \sum_{k = 1}^{\infty } \frac{x_{k}}{3^{k}}$;

$X = L_{p}(\left[0,1\right]), f(x) = \int_{0}^{\frac{1}{2}} x(t) d t-\int_{\frac{2}{3}}^{1} x(t) d t$.

$X = L_{p}([1,+\infty )), f(x) = \int_{1}^{+\infty } \frac{x(t)}{t^{\alpha }} d t$;

$X = l_{p}, f(x) = \sum_{n = 1}^{\infty } \alpha^{n} x_{n}$.