1) Cauchy-Schwarz inequality states that the absolute value of vector inner product is always less or equal to product of norms of individual vectors i.e., $|a^Tb|\leq\Vert a\Vert_2 \Vert b\Vert_2$.
Does this inequality hold true for any Lp vector norms especially the L1 norm i.e., $|a^Tb|\leq\Vert a\Vert_1 \Vert b\Vert_1$.? If yes any Proofs?
2) Does this Cauchy-Schwarz inequality hold for a matrix $A$ having linearly independent columns $|A^Tb|\leq\Vert A\Vert_F \Vert b\Vert_2$