Is any norm induced by some inner product?

Question

It is a well-know fact that an inner product induces some norm.

How about the converse? I think it's false but I can't think of an example.

I'm thinking some properties like the parallelogram law will not hold with any norm...

Answer 1

Good thought with the parallelogram law. In fact, a norm will satisfy the parallelogram law if and only if it can be induced by some inner product. Given a norm satisfying the parallelogram law, we can recover the inner product inducing that norm using the polarization identity.

Some common examples: the $p-$norm given by $\|x\|_p = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}$) is a norm for any $p \geq 1$, but will only satisfy the parallelogram law for $p=2$.

Another important example of a norm that doesn't obey the parallelogram law is the "$\infty$-norm" given by $\|x\|_{\infty} = \max_i\{|x_i|\}$.

Answer 2

In inner product spaces, you have the equality $$||x+y||^2+||x-y||^2 = 2(||x||^2+||y||^2).$$ This equality can easily be shown by writing $\langle x,x\rangle$ for $||x||^2$.

In general, this does not hold. See the space $\mathbb R^2$ with the norm $||(x,y)||=\max\{|x|,|y|\}$.

Answer 3

Of course, the norm on $\mathbb{R}^2$ given by $\|(x,y)\|=(|x|^{3.63421}+|y|^{3.63421})^{1/3.63421}$, where $|{\cdot}|$ denotes the absolute value, is not induced by any inner product, still it is a norm.