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I've read in the solution of an exercise: "$T$ has a finite norm, thus $T$ is continuous".

We are in a normed vector space $(V,||.||)$ and $T$ is a linear selfmap over the vector space $V$. The purpose of the exercise is to show that the set $L_w=\{v\in V:T(v)=w\}$ is closed. The first statement in the solution is the one above, and once showed continuity of $T$, the exercise is pretty straightforward.

Since I am working with linear operators in normed vector spaces, my idea is that it is linked with the equivalence of boundedness and continuity of an operator, but I am not sure. Is my idea right? If not, why the statement is true?

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The norm of a linear operator $L:V\to V$ is the smallest $M$ such that:

$$ ||Lv||\le M||v|| \quad \forall v \in V $$

So, your idea is correct: a linear operator $L$ in a normed spaces $V$ is bounded if and only if it is a continuous linear operator (here a simple proof).

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