Let $X$ be a topological vector space over $\mathbb R$ or $\mathbb C$. A subset $B\subset X$ is defined to be bounded if
for any open neighborhood $N$ of $0$ there is a number $\lambda>0$ such that $B\subset \mu N$ for any $\mu>\lambda$.
I was wondering whether this notion of boundedness is equivalent to saying that for any open neighborhood $N$ of $0$ there is a $\mu>0$ such that $B\subset \mu N.$
By definition $\mu N:=\{\mu\cdot x\colon x\in N\}$.