Math Problems and solutions
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19.5.1 Compactness
3.75 $
19.5.2 Compactness
19.5.3 Compactness
4.99 $
![Problem: Is the set of functions \[ M=\{x \in C([0,1]):|x(t)| \leq t \forall t \in[0,1]\} \] precompact in the space \( C([0,1]) \) ?](/storage/uploads/task_mls/765/en/19.5.1.png){kind=link}
![Problem: Prove that in the space \( C[0,2] \) the following set is precompact \[ A=\left\{f \in C[0,2]|| f^{\prime \prime}(x)|\leq 3, \forall x \in[0,2] ;| f^{\prime}(0) \mid \leq 1, f(2)=3\right\} \text {. } \]](/storage/uploads/task_mls/1466/en/19.5.2.png){kind=link}
![Problem: Prove that in the space \( C[-1 ; 2] \) relatively compact (precompact) set \[ E=\left\{\varphi \in C[-1 ; 2],\left|\int_{-1}^{2} x \varphi(x) d x\right| \leq 3,\left|\varphi^{\prime}(x)\right| \leq 3\right\} . \]](/storage/uploads/task_mls/1467/en/19.5.3.png){kind=link}