If the inner product induces the l2 norm, what kind of non-inner product induces a general Lp norm where $p \neq 2$?

$(X, ||•||)$ be normed space.

Then the norm $||•||$ induce an inner product on the linear space $X$ iff it satisfy the parallelogram law

$$||x+y||^2 +||x-y||^2 =2(||x||^2 +||y||^2 ) $$

Then the inner product can be given explicitly by polarization identitiy.

$\langle x, y \rangle =\frac {1}{4} { \sum_{k=0}^{3} {i^k ||x+i^k y ||^2}}$

Now you can check when $L_p$ norm induce an inner product on the underlying linear space.

An inner product on a linear space is the generalization of dot product of $\mathbb{R^2} $ or $\mathbb{R^2}$. It gives us lots of advantage. In an inner product space one can talk about the angle between vectors and length of vectors.

$L_p$ norm for $p\neq 2 $ is just a norm, it doesn't represent an inner product.