Math Problems and solutions
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19.1.2.1 Convergence in metric spaces
1.43 $
19.1.2.2 Convergence in metric spaces
4.28 $
19.1.2.3 Convergence in metric spaces
2.38 $
19.1.2.4 Convergence in metric spaces
2.85 $
![Problem: Investigate the convergence of the sequence of functions in space: \[ f_{n}(x)=\sqrt{n}(1-x)^{n} \text { in } L_{2}[0,1] \text {. } \]](/storage/uploads/task_mls/804/en/19.1.2.1.png){kind=link}
![Problem: Will the function \( d(x, y) \) \( =\int_{0}^{1} \frac{|x(t)-y(t)|}{e^{t}} d t \) set a metric on \( C([0,1]) \) ? If this function defines a metric, will the sequence \( \left\{x_{n}\right\} \), defined by the formula, \( x_{n}(t)=\left\{\begin{array}{ll}n t, & t \in\left[0 ; \frac{1}{n}\right] \\ 1, & t \in\left[\frac{1}{n} ; 1\right]\end{array}\right. \) be 1) Fundamental; 2) Converging relative to the given metric?](/storage/uploads/task_mls/805/en/19.1.2.2.png){kind=link}
![Problem: Does the sequence of the function \( f_{n}(x)= \) \( n \sqrt{x^{n}}(1-\sqrt{x})^{n} \) converge on the space \( C([0,1]) \) ?](/storage/uploads/task_mls/806/en/19.1.2.3.png){kind=link}
![Problem: Does the sequence of elements converge on the space \( l_{1} \) ? \[ x_{n}=\left(\frac{1-n^{2}}{n^{2}}, \frac{2-n^{2}}{n^{2}}, \ldots, \frac{n-n^{2}}{n^{2}}, 0,0, \ldots, 0, \ldots\right) \text {. } \]](/storage/uploads/task_mls/807/en/19.1.2.4.png){kind=link}