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Problem: Prove that an open ball is an open set, and a closed ball is a closed set.

19.1.1.8 Properties of metric spaces

2.54 $

Problem: Bring an example of a fundamental sequence on the space $ (0,1) $ with the metric $ \rho(x, y)=\left|\frac{1}{\sin x}-\frac{1}{\sin y}\right| $, which is not convergent.

19.1.1.9 Properties of metric spaces

2.54 $

Problem: Let $ X $ be an $ \mathrm{n} $-dimensional space. Is $ \left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|\left(x=\left(x_{1}, \ldots, x_{n}\right) \in X\right) $ a norm on $ X $ ?

19.2.1.1 Properties of normed spaces

2.54 $

Problem: Will the functions \[ \|\mathrm{x}\|_{1}=\int_{0}^{1}|x(t)| d t \text { and }\|x\|_{2}=\int_{0}^{1} e^{t}|x(t)| d t \] be norms on the space $ C([0,1]) $ ? If yes, will these norms be equivalent on the given space?

19.2.1.2 Properties of normed spaces

2.54 $

Problem: Will $ \|x\|=2\left|x_{1}\right|+3\left|x_{2}\right|\left(x=\left(x_{1}, x_{2}\right)\right) $ be a norm on $ \mathbb{R}^{2} $ ?

19.2.1.3 Properties of normed spaces

2.54 $

Problem: Is the mentioned function $ \rho(x, y) $ a metric in the given set? Fill in the table. \begin{tabular}{|c|c|c|c|c|c|} \hline Set & \begin{tabular}{l} $ \quad \mathbb{R} $ \\ Of real \\ numbers \\ $ x, y $ \end{tabular} & \begin{tabular}{l} $ \mathbb{R}^{3} $ \\ Of 3 dimension vectors \\ $ x=\left(x_{1}, x_{2}, x_{3}\right) $ \\ $ y=\left(y_{1}, y_{2}, y_{3}\right. $ \end{tabular} & \begin{tabular}{l} $ \qquad \quad S(\mathbb{R}) $ \\ Of all possible \\ sequences of real \\ numbers \\ $ x=\left(x_{1}, x_{2}, \ldots, x_{n}, \ldots\right) $ \\ $ y=\left(y_{1}, y_{2}, \ldots, y_{n}, \ldots\right) $ \end{tabular} & \begin{tabular}{l} $ \qquad F_{c}([a ; b], \mathbb{R}) $ \\ Of countinuous \\ functions \\ $ x(t):[a ; b] \rightarrow \mathbb{R} $ \\ $ y(t):[a ; b] \rightarrow \mathbb{R} $ \end{tabular} & \begin{tabular}{l} $ \qquad M $ \\ Of the cells $ x, y $ of \\ the chessboard $ 8 \times $ \\ 8 \end{tabular} \\ \hline Function $ \rho(x, y)=\cdots $ & $ \cos |x-y| $ & $ \sqrt{\left(x_{1}-y_{1}\right)^{2}+\left(x_{2}-y_{2}\right)^{2}} $ & $ \sup _{n \in \mathbb{N}}\left|x_{n}-y_{n}\right| $ & $ \int_{a}^{b}|y(t)-x(t)| d t $ & \begin{tabular}{l} The minimum \\ number of the \\ knight moves on the \\ way from $ x $ to $ y $ \end{tabular} \\ \hline Colclusion: & & & & & \\ \hline \begin{tabular}{l} Function $ \rho $ doesn't set a metric in the given set, since it's \\ not defined for all pairs $ (x, y) $ \end{tabular} & & & & & \\ \hline \begin{tabular}{l} Function $ \rho $ doesn't set a metric in the given set, since it \\ isn't non-negative \end{tabular} & & & & & \\ \hline \begin{tabular}{l} Function $ \rho $ doesn't set a metric in the given set, since it \\ isn't non-degenerate \end{tabular} & & & & & \\ \hline \begin{tabular}{l} Function $ \rho $ doesn't set a metric in the given set, since it \\ doesn't have symmetry property \end{tabular} & & & & & \\ \hline \begin{tabular}{l} Function $ \rho $ doesn't set a metric on the given set, since it \\ doesn't satisfy the triangle inequality \end{tabular} & & & & & \\ \hline Function $ \rho $ doesn't set a metric on the given set. & & & & & \\ \hline \end{tabular}

19.1.1.2 Properties of metric spaces

12.7 $

Problem: Is the sequence $ x_{n}=(\underbrace{\frac{1}{n}, \frac{1}{n}, \ldots, \frac{1}{n}}_{n^{2}}, 0,0, \ldots) $ converging on the space $ l_{2} $ ?

19.2.2.1 Convergence in normed spaces

2.54 $

Problem: The scale of $ l_{p} $ spaces. Prove that $ \lim _{p \rightarrow \infty}\|x\|_{l_{p}}=\|x\|_{l_{\infty}} $.

19.2.2.2 Convergence in normed spaces

5.08 $

Problem: Find out if the sequence \[ x_{n}=\left(1, \frac{1}{\sqrt{2}}, \ldots, \frac{1}{\sqrt{n}}, 0, \ldots\right) \] converges in the normed space $ l_{2} $.

19.2.2.3 Convergence in normed spaces

3.81 $

Problem: Is the sequence $ x_{n}(t)=e^{\frac{t}{n}} $ converging on the space $ C_{2}[0 ; 1] $ ?

19.2.2.4 Convergence in normed spaces

3.05 $

Problem: Find out the character of the convergence of the sequence of the function: \[ f_{n}(x)=x^{n}-x^{2 n}, \quad g_{n}(x)=x^{n}-x^{n+1} \] on the segment $ [0 ; 1] $.

19.2.2.5 Convergence in normed spaces

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Problem: Find out the character of the convergence of the sequence of the function: \[ f_{n}(x)=\operatorname{tg}^{-1} \frac{\pi}{2 n} x \] a) on the set $ \mathbb{R}, b) $ on the segment $ [0 ; 1] $.

19.2.2.6 Convergence in normed spaces

3.81 $

Problem: Find out the character of the convergence of the sequence of the function: $ f_{n}(x)=\frac{n x}{n^{2}+x^{2}} $ on the set $ \mathbb{R} $.

19.2.2.7 Convergence in normed spaces

2.54 $

Problem: On $ C^{1}[0 ; 1] $ the following norms are given: \[ \begin{array}{l} \|f\|_{1}=|f(0)|+\int_{0}^{1} t^{2}\left|f^{\prime}(t)\right| d t \\ \|f\|_{2}=\int_{0}^{1}|f(t)| d t+\int_{0}^{1} t^{2}\left|f^{\prime}(t)\right| d t . \end{array} \] Is it true that one of these norms is stronger than the other one? Are they equivalent?

19.2.1.4 Properties of normed spaces

6.35 $

T Problem: Calculate the line integral of the 2nd kind using Green's formula. \[ \oint_{C}\left(x^{3}-2 y\right) d x+\left(y^{3}-x\right) d y \] where $ C: x=1, y^{2}=x $.

9.4.11 Curvilinear integrals

1.27 $