Convergence in normed spaces

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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

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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

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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

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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

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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

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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

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Problem: Find the norm of \( x(t)=\sin ^{3} t \) on \( C_{1}[0 ; 1] \).

19.2.2.8 Convergence in normed spaces

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Problem: Find the limit of the sequence in the space \( C^{1}[0 ; 1] \) : \[ f_{n}(x)=\int_{0}^{1} x\left(t^{n}\right) d t \]

19.2.2.9 Convergence in normed spaces

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