Does there exist a sequence on the unit sphere which converges to zero under compact operator? Let  $X$ be an infinite dimensional Banach space and $T: X \rightarrow Y$ be compact operator (between Banach spaces).
Question: Does there always exist a sequence $x_n$ of norm $1$ such that $T(x_n)$ converges to zero?
If we assume $X$ to be reflexive, this is the case: Choose a sequence on the unit sphere which converges weakly to zero. The compact operator $T$ then maps this sequence to a sequence converging to zero.

If your claim were false, $T$ would need to be injective.
Furthermore, it would follow that the constant
$$
C := \inf_{\|x\|=1} \|Tx\|
$$
is positive. From this, it follows that $T(X)$ is closed and $T : X \to T(X)$ is a bijective linear map between Banach spaces and thus an isomorphism of Banach spaces.
But $T(B_1 (0))$ is precompact, so that $B_1 (0)$ is itself precompact. It follows that $X$ is finite dimensional, a contradiction.
Hence, we can always find a sequence with the desired properties.
