difference between normed linear space and inner product space

I've seen that the definitions of normed linear space and inner product space for a complex vector space $V$ are very close to each other except for the fact that one is defined on $V$ and the other on $V\times V.$ The properties are almost the same. I don't understand what are the differences except for the fact mentioned above. Do both of them have their own importance in the context that both of them provide a properties that are not obtainable by studying the other?

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Every inner product $\langle,\rangle$ gives rise to a norm defined by $\|x\|^2=\langle x,x\rangle$. But a norm in a normed space does not necessarily arise from an inner product. There are normed spaces whose norm can't be induced in the way above from any inner product. So, the class of normed spaces is truly larger than the class of all (normed spaces induced by the inner product in) inner product spaces.

In fact, there are several conditions on the norm in a normed space that guarantee that the norm is induced by an inner product. See this article for instance.

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If you have an inner product space $\left(E, \varphi\right)$, it has a natural structure as a normed vector space: $\left(E,x\mapsto \sqrt{\varphi(x,x)}\right)$ but the other way around isn't true. There are norms that do not come from inner products.

And example with $E=\Bbb R^2$

If you take $\varphi:\left(\left(x_1,y_1\right),\left(x_2,y_2\right)\right)\mapsto x_1x_2+y_1y_2$ you have an inner product.

And if you let $N_2:\left(x,y\right) \mapsto \sqrt{\varphi\left(\left(x,y\right),\left(x,y\right)\right)}=\sqrt{x^2+y^2}$, you get the norm you know.

But there are other norms such as $N_\infty:(x,y)\mapsto \max(x,y)$ that can't be built from an inner product.

By the way, if your norm $N$ does come from an inner product, you can get the inner product back by letting $\psi:(x,y)\mapsto \cfrac{N(x+y) ^2-N(x-y)^2}{4}$

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