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The question is as follows:

Show that a linear operator $T:X\to Y$ where $X$ and $Y$ are normed spaces is compact if and only if the image $T(M)$ of the unit ball $M\subset X$ is relatively compact in $Y$.

The forward part of the proof is simple because a unit ball is a bounded subset of $X$ and $T$ is compact and hence it maps the bounded subset of $X$ to a relatively compact subset $T(M)$ of $Y$.

However i am slightly confused about how to proceed with proving the converse part.

If the image $T(M)$ of the unit ball $M\subset X$ is relatively compact in $Y$ is true then how can this be extended to cover every possible bounded subset of $X$ to test $T$ for compactness ?

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Hint: since the operator $T$ is linear you can scale and translate any bounded set to be contained in the unit ball.

If $T$ in not linear checking relative compactness only on the unit ball is generally not enough.

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