91影视

The first case is: $$ V = \{f \in C(鈩�) \mid \lim_{x \rightarrow \pm \infty} f(x) = 0\} $$ with $$ \|f\| = \max_{x \in 鈩潁|f(x)| $$ We will check whether the given norm satisfies all the properties of a norm on V. 1. Positivity: Clearly, \({\|f\|}_{|f(x)| \ge 0}\) for all \(f \in V\). Moreover, \({\|f\|}_{} = 0 \) if \(f(x) = 0\) for all \(x \in 鈩漒). 2. Linearity: Let \(伪 鈭� 鈩俓) and \(f \in V\). $$ \|伪f\| = \max_{x \in 鈩潁 |伪f(x)| = |伪| \max_{x \in 鈩潁 |f(x)| = |伪| \|f\| $$ 3. Triangle inequality: Let \(f, g \in V\). $$ \|f + g\| = \max_{x \in 鈩潁 |f(x) + g(x)| \leq \max_{x \in 鈩潁 (|f(x)| + |g(x)|) \leq \max_{x \in 鈩潁 |f(x)| + \max_{x \in 鈩潁 |g(x)| = \|f\| + \|g\| $$ Since the given norm satisfies all the properties of a norm, the space V is a normed space for case (a). Proceeding with the second case:

The second case is: $$ V = \{(a_n) 鈭� 鈩� \mid \text{sequence } (a_n) \text{ converges}\} $$ with $$ \|(a_n)\| = \left|\lim_{n \rightarrow \infty} a_n\right| $$ Let's check if these norms fulfill the properties of norms on V. 1. Positivity: Clearly, \({\|\left(a_{n}\right)\|}_{|a_{n}| \ge 0}\) for all \({(a_{n}) \in V}\). However, the norm \({\|\left(a_{n}\right)\|}_{} = 0 \) does not imply \({a_{n}} = 0 \) for all \(n \in {鈩晑\), since \({(a_{n}) 鈭� V}\) only requires the sequence \({(a_{n})}\) to converge, not to converge to \(0\). Hence, the given norm does not satisfy the property of Positivity, and so case (b) cannot be a normed space. Proceeding with the third case:

The third case is: $$ V = \{(a_n) 鈭� 鈩� \mid \text{sequence } (a_n) \text{ is a null sequence}\} $$ with $$ \|(a_n)\| = \max_{n 鈭� 鈩晑 |a_{n}| $$ Let's check whether these norms satisfy the properties of norms on V. 1. Positivity: Clearly, \({\|\left(a_{n}\right)\|}_{|a_{n}| \ge 0}\) for all \({(a_{n}) \in V}\). Also, \({\|\left(a_{n}\right)\|}_{} = 0 \) implies \({a_{n}} = 0 \) for all \(n \in {鈩晑\) since \({(a_{n}) 鈭� V}\) requires the sequence \({(a_{n})}\) to be a null sequence. 2. Linearity: Let \(伪 鈭� 鈩俓) and \({(a_n) 鈭� V}\). $$ \|伪(a_n)\| = \max_{n 鈭� 鈩晑 |伪a_n| = |伪| \max_{n 鈭� 鈩晑 |a_n| = |伪| \|(a_n)\| $$ 3. Triangle inequality: Let \({(a_n), (b_n) 鈭� V}\). $$ \|(a_n + b_n)\| = \max_{n 鈭� 鈩晑 |a_n + b_n| \leq \max_{n 鈭� 鈩晑 (|a_n| + |b_n|) \leq \max_{n 鈭� 鈩晑 |a_n| + \max_{n 鈭� 鈩晑 |b_n| = \|(a_n)\| + \|(b_n)\| $$ Since the given norm satisfies all the properties of a norm, the space V is a normed space for case (c).