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I tried assuming [tex]f_1 \otimes f_2[/tex] acting on a general element [tex]\sum v_1 \otimes v_2[/tex] was zero, but the resulting tensor [tex]\sum f_1(v_1) \otimes f_2(v_2)[/tex] is too complicated for me to draw implications for [tex]v_1[/tex] and [tex]v_2[/tex]. It is obvious that [tex]v_1 \in \ker f_1[/tex] or [tex]v_2 \in \ker f_2[/tex] implies that the latter tensor product is 0, but what can be said for the other direction?