# $B\subset E$ is bounded $\iff$ $\forall b_n\in B, \forall \lambda_n\in \mathbb R$ such that $\lambda_n\to 0$ then $\lambda_n b_n\to 0$

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.

• Why is the answer not accepted? – 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, $$\lambda_n b_n \not\to 0$$.)