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I'm currently reading up on Banach spaces. My book "Introduction to Banach spaces and their Geometry" by Beauzamy mentions "normed spaces" in some places, and "normed linear spaces" in other. I really don't understand the difference.

On searching on the internet, I found that the same axioms apply to both.

Is the only difference that normed linear spaces are to do with geometrical linear spaces, while normed spaces are to do with a larger class of vector spaces?

Thanks in advance!

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I don't have the book you mentioned, but, as far as I know, they are the same thing. You should look for definitions in the book, but my understanding is as follows:

A norm is a function with certain properties that is defined on a vector space; the vector space is then said to be a "normed vector space". The defining properties of a norm function refer to algebraic operations (addition and scalar multiplication) that exist only in a vector space. So it does not make sense to try to define a norm on a space that is not a vector space.

So, every "normed space" is a normed vector space.

In the above, I used the term "vector space" instead of "linear space"; they mean the same thing.

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