Properties of metric spaces

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Problem: Is $ \rho(x, y)=\max _{1 \leq k \leq n} k\left(x_{k}-y_{k}\right) $ a metric on $ \mathbb{R}^{n} $, where $ x, y \in \mathbb{R}^{n}, x=\left(x_{1}, \ldots, x_{n}\right), y=\left(y_{1}, \ldots, y_{n}\right) $ ?

19.1.1.1 Properties of metric spaces

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Problem: Let the metric on $ \mathbb{R} $ be $ \rho(x, y)=\mid \tan ^{-1} x- $ $ -\tan ^{-1} y \mid $. Prove that the obtained space is not complete. Indication: consider the sequence $ x_{n}=n $.

19.1.1.4 Properties of metric spaces

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Problem: The metric $ \rho(x, y)=\tan ^{-1}|x-y| $ is given on $ \mathbb{R} $. Find out if this space is complete.

19.1.1.5 Properties of metric spaces

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Problem: The metric $ \rho(x, y)=\left|x^{3}-y^{3}\right| $ is given on $ \mathbb{R} $. Find out if this space is complete.

19.1.1.6 Properties of metric spaces

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Problem: Prove that the set of rational numbers is not complete on space $ \mathbb{R} $.

19.1.1.7 Properties of metric spaces

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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

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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

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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

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Problem: Let $ x^{0}, x^{1} \in l_{1} $. Find $ \varepsilon- $ neighbourhood of the point $ x^{0} $, where the point $ x^{1} $ lies. \[ l_{1}=\left\{x=\left(x_{1}, \ldots, x_{n}, \ldots, \ldots\right)\left|\sum_{n=1}^{\infty}\right| x_{n}\left|<\infty, \rho_{1}(x, y)=\sum_{n=1}^{\infty}\right| x_{n}-y_{n} \mid\right\}, x^{0}=(0, \ldots, 0, \ldots), x^{1}=\left(1, \frac{1}{4}, \ldots, \frac{1}{n^{2}}, \ldots\right) \text {. } \]

19.1.1.10 Properties of metric spaces

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