what is the difference between norm and modulus?

[cks]

norm is defined to be the length of the vector and we put we denote it by ||a||.

However, modulus |a| also means the length of a from the origin?

So, what is the difference between the symbol || || and | |?

[quasar987]

There is none. Some authors use one, others use the other.

[daveb]

I was always taught that the norm is defined to be any length which satisfies the definition of a norm, but modulus is specifically the Euclidean norm. That could have just been the instructor/book, though I suppose.

[dextercioby]

The modulus is the norm on \mathbb{R} . \left(\mathbb{R},\left| , \right| \right) is a Banach space.

[cks]

I think norm is the length of a point from the origin.

whereas the modulus is more of a distance from one point to another point.

norm is just a specific case of the distance from a point to its origin.

[D H]

The semi-standard usage is that modulus is specialized to the reals (absolute value), complex numbers (complex modulus), and quaternions. However, some write |x| instead of ||x|| to mean norm. Norm is a very generalized concept that covers everything from Euclidean distance to distance as measured on a road grid (taxicab norm) to the L-infinity norm, and beyond. Anything that qualifies as a distance can be used as a norm. Even things much more complex and abstract than vectors can have a norm. Well-defined norms exist for matrices, for example.

[Chris Hillman]

Actually, both norm and modulus are potentially ambiguous terms in mathematics; the intended meaning depends upon context. Generally speaking, modulus often suggests algebraico-geometric-analytic origins (as in the theory of elliptic functions) while norm tends to suggest operator algebras and functional analysis. In the context of functional analysis, esp. Banach spaces, I agree with DH about what one can reasonably expect "modulus" to denote.