Math Problems and solutions
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19.2.1.1 Properties of normed spaces
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19.2.1.2 Properties of normed spaces
19.2.1.3 Properties of normed spaces
19.2.1.4 Properties of normed spaces
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19.2.1.5 Properties of normed spaces
![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?](/storage/uploads/task_mls/722/en/19.2.1.2.png){kind=link}
![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?](/storage/uploads/task_mls/732/en/19.2.1.4.png){kind=link}
![Problem: Prove that the Apollonius identity holds in Euclidean space: \[ 2\|z-x\|^{2}+2\|z-y\|^{2}=\|x-y\|^{2}+4\left\|z-\frac{x+y}{2}\right\|^{2} . \]](/storage/uploads/task_mls/1469/en/19.2.1.5.png){kind=link}