Is there a bounded dense subset of norm linear space?

Question

I have a question. In norm linear space $X$, we can find a bounded dense subset of $X$, can´t we?

Answer 1

You can only if $X$ is bounded itself. This is because of the continuity of the norm.

Consider a bounded dense part $B$, 2 sequences $x_n,y_n \in B$ such as $x_n\to x, y_n\to y$ and such as, $D$ being the diameter of $X$, $\|x-y\| > D- \epsilon$. Let $D_B$ be the diameter of $B$.

$$ D_B\ge \|x_n-y_n\| \to \|x-y\| > D-\epsilon $$ using the continuity of the norm, which implies $D_B \ge D$ (and as $B\subset X$, $D_B = D$).

Hence $D_B < \infty \implies D<\infty$.

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NB: continuity of the norm: $$ f = \|.\|:X\to \Bbb R^+\\ |f(x) - f(y)| \le \|x - y\| $$is the triangle inequality.