# Proving that $T$ is open implies that $0$ is an interior point of $\overline{T(B_X)}.$

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:

• 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. – s.harp Mar 25 at 18:28
• @s.harp can you give me more details about the idea of the backward direction please? – Idonotknow Mar 26 at 4:56