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I'm currently working my way through the book "Linear Algebra Done Right" by Sheldon Axler.

In the book, he defines a norm the following way.

Let $V$ be an inner product space and let $\langle u, v\rangle$ denote the inner product of $u$ and $v$. Then, the Norm of any vector $v \in V$ is defined as $||v|| = \sqrt{\langle v, v\rangle}$.

However, I was digging through Wikipedia and I discovered the concept of a "Normed Vector Space", i.e. a vector space on which a norm is defined. Inner product spaces seemingly constitute a subset of normed vector spaces.

My question is this - how is it possible to have a normed vector space that is not an inner product space if it seems like the definition of a norm depends on the presence of an inner product?

Noah Stebbins
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  • The definition of the norm of a normed space does not depend on an inner product. You can find it here. Given an inner product space you can define a norm the way you quoted; that is, a function satisfying the definition of the norm of a normed space. – posilon Nov 21 '20 at 21:53
  • This may also be of interest. – posilon Nov 21 '20 at 21:57
  • Look at https://math.stackexchange.com/questions/159766/an-example-of-a-norm-which-cant-be-generated-by-an-inner-product – KCd Nov 21 '20 at 21:58

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Not quite. If we have an inner product, ee can use it to define a norm. But there are norms that are induced by no inner product. For instance,$$\|(x,y)\|=|x|+|y|.$$

José Carlos Santos
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