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Problem list Free problems

Attention! If a subsection is selected, then the search will be performed in it!

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: 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 $

Problem: Prove that the Apollonius identity holds in Euclidean space: \[ 2\|z-x\|^{2}+2\|z-y\|^{2}=\|x-y\|^{2}+4\left\|z-\frac{x+y}{2}\right\|^{2} . \]

19.2.1.5 Properties of normed spaces

2.54 $

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