Here is the question:

Let $X,Y$ be Banach spaces and let $T \in \mathcal{L}(X, Y). $ Prove that $T$ is open if and only if $0$ is an interior point of $\overline{T(B_X)}$.

Where $B_X$ is the open unit ball in $X.$

My questions are:

1-Any hints about proving the forward direction?

2-Is this what is called the "open unit ball lemma" page 286 in Kreyszig "Introductory Functional Analysis with Applications"?

3- what does this problem want to teach us?

Here is the lemma in Kreyszig I am referring to:

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  • 1
    $\begingroup$ Well $B_X$ is open, right? And if $T$ is open then $T(B_X)$ is open, containing $0$. For the other direction, note that rescaled and translated open balls form a base of the topology. Meaning a set is open every point in it has an open ball around it contained in the set. Thats the important step for the other direction. $\endgroup$ – s.harp Mar 25 at 18:28
  • $\begingroup$ @s.harp can you give me more details about the idea of the backward direction please? $\endgroup$ – Idonotknow Mar 26 at 4:56

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