"Minkowski's inequality" taking on negative functions in $L^p$ confusion

Question

Does the "Minkowski inequality" imply that

$||f_n-f||_p=||f_n+-f||_p \leq ||f_n||_p + ||-f||_p = ||f_n||_p -||f||_p?$

Answer 1

No, it does not.

The problem is in the last "$=$" sign. In particular, $\|-f\|_p = -\|f\|_p$ is false, if $f\neq0$. Instead, $$ \|-f\|_p = \|f\|_p $$ is correct.

Answer 2

Norms are absolutely homogeneous, meaning that $\lVert \alpha f \rVert = |\alpha| \lVert f \rVert$ for $\alpha \in \mathbb{R}.$ So your last equality doesn't follow. In fact $\lVert -f \rVert = \rVert f \lVert.$