# Difference: normed space and normed linear space.

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?

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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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