Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.