Equivalent norms and orthogonality

Question

Is orthogonality transferred by equivalent norms in any $L^p$ space? it looks so since orthogonality is defined in terms of norms.

We say that two norm are equivalent is there exists numbers $C$ and $D$ such that for all $x$

$\mid\mid x \mid\mid_{b} C \le \mid\mid x \mid\mid_{a} \le D \mid\mid x \mid\mid_{b} $

Answer 1

The answer to your question is yes in general Banach space because if $\Vert\cdot\Vert_a \sim \Vert \cdot \Vert_b$ are equivalent norms, then $$(V,\Vert\cdot\Vert_a )^* = (V,\Vert\cdot\Vert_b)^* $$ so clearly for a subspace $W \leq V$ we have $$W^{\perp,a} = W^{\perp,b} \leq V^*$$