Two ways to express boundedness

Question

I'm little confused about the boundedness in a Banach space. Here are two boundedness definition we can encounter in Banach space:

1) A set $E$ is bounded if, for every neighborhood of 0, we have $E \subset tV$ for all sufficiently large $t$.
2) A set $E$ is bounded if there exists $M > 0$ that $||x|| \le M$ for all $x \in E$

Are the two definition the same in Banach space. If not, which one implies the other. I hope someone can help me clarify this. Thanks.

Answer 1

The first definition makes sense in any topological vector space; the second requires a norm. In any normed linear space they are equivalent. Suppose first that $\|x\|\le M$ for all $x\in E$, and let $V$ be any nbhd of $0$. Then there is some $\epsilon>0$ such that $V\supseteq B(0,\epsilon)$, and clearly

$$E\subseteq B(0,M+1)=\frac{M+1}\epsilon B(0,\epsilon)\subseteq tV$$

for all $t\ge\frac{M+1}\epsilon$.

Now suppose that for each $t\in\Bbb R^+$ there is an $x_n\in E$ such that $\|x_t\|>t$, and let $t>0$ be arbitrary:

$$x_t\notin B(0,t)=tB(0,1)\;,$$

so there is no $t>0$ such that $E\subseteq tB(0,1)$, but $B(0,1)$ is of course a nbhd of $0$.