Let $E$ be a normed space. Then $B\subset E$ is bounded $\iff$ $\forall b_n\in B, \forall \lambda_n\in \mathbb R$ such that $\lambda_n\to 0$ then $\lambda_n b_n\to 0$

One of definitions of the boundedness of a normed space:

$B$ is bounded $\iff$ $\exists \rho>0$ such that $||b||<\rho,\; \forall b\in B$

Using definition $\Rightarrow$ side is easy to prove as following:

Right side:$\Rightarrow$

$\Rightarrow$ : Let $E$ be a normed space and assume $B\subset E$ is bounded and let $b_n\in B$ and $\lambda_n\in R,\; \lambda_n\to 0$ and $\epsilon>0$ are given arbitrarly.

Therefore, $\exists \rho>0$ such that $||x||<\rho,\; \forall x\in B$,

and in particular $||b_n||<\rho, \forall n$ and since $\lambda_n\to 0$ $\exists N\in \mathbb N$ such that $\forall n\ge N$, $|\lambda_n|<\epsilon$.

So $||\lambda_n b_n||=|\lambda_n|||b_n||<|\lambda_n|\rho <\epsilon$.

About Left side $\Leftarrow$: If $B$ were a field I can find an example $b_n=n^2$ and $\lambda_n=1/n$ suchthat $|\lambda_n b_n|=|n|\to \infty$ it proves the contrapositive of the $\Leftarrow$ side but $B$ is a normed space not a field so I thought that maybe I can think some homomorphisms between $B$ and $\mathbb R$ to do this.(Question 1: Is there something like that we can make connection between normed linear spaces and fields(in a vector space manner))

Question 2(Main one): How can one prove $\Leftarrow$ properly.

  • $\begingroup$ Why is the answer not accepted? $\endgroup$ – Ramanujan Dec 28 '20 at 22:49

Suppose $B$ is not bounded, then there is a sequence $b_n$ with $\|b_n\| \to \infty$ in $B$. Then $\lambda_n = {1 \over \|b_n\|} \to 0$, and $|\lambda_n b_n| = 1 \to 1$. (In particular, $\lambda_n b_n \not\to 0$.)


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