连续线性函数在单位球面恒为正,则下确界为正

hbghlyj

A2-metricspaces.pdf Lemma 2.2.1说明:
Let $f : V → W$ be a linear map between normed vector spaces. Then
1) $f$在0连续, 则$\{‖f (x)‖ : ‖x‖≤r\}$有界, 对任何$r>0$.
2) 若存在$r$使$\{‖f (x)‖ : ‖x‖=r\}$有界, 则$f$在$V$上一致连续.
又见Bounded operator
我想问一个问题:
Let $f : V → ℝ$ be a continous linear map between normed vector spaces. By the above, $\inf f(S)$ exists.
If $f>0$ for all $S=\{x∈V:‖x‖=1\}$, then $\inf f(S)>0$.
这个正确吗? (Note that $S$ is not necessarily compact)

hbghlyj

Czhang271828

f>0 啥意思，值域都不是 R 啊  发表于 2022-11-13 14:43

哦对! 是我打错了. 已修改