Question

am i right in saying it only makes sense to define the norm of an inner product space, rather than the norm of a vector space?

Answer 1

u DO NOT define the norm of an inner product. u DO define the norm of a vector.

Answer 2

the norm is a function on an inner product space, right?

Answer 3

in defining the norm of a vector u can use an inner product:
\[ \large ||v||=\sqrt{(v\mid v)} \]
or not.

a norm is a function with domain a vector space.

Answer 4

is it possible to have more than one inner product on a vector space?

Answer 5

yes, of course.

Answer 6

for instance, in R^n u can have infinitely many inner products
\[ \large (x\mid y)=xAy^t \]
where A has to be a positive semi-definite matrix.

Answer 7

that's what i thought. and that's why i think there might be a flaw in the definition im given, or (more likely) im being stupid.

this is the definition i've been given:

Let (−|−) be an inner product on a real vector space V . If v ∈ V then the
norm of v is
\[||v|| = \sqrt{(v|v)}\]



now, surely here norm(v) is not well defined as there are many possible inner products. or are we claiming that they all take the same value when put in this formula?

Answer 8

on the contrary. since u can have several inner products on a given, each one of them in turn defines a different norm.

so. there's no such thing of THE norm of a vector. rather THE norm RELATIVE to a given inner product (assuming u used one to define the norm).

Answer 9

ah! this makes sense. thanks so much for your help :D

Answer 10

but again. a (real) norm is a function \(f:V\to\mathbb{R}\) that satifies the properties of the absolute value.

Answer 11

u r welcome

Answer 12

the function (norm) need not come from an inner product.

i recomend u "Linear Algebra" by Hoffman and Kunze

Answer 13

yeah, our lecture course has approached it from the inner product perspective

i have "Linear Algebra" by Kaye and Wilson which seems to explain both approaches. i was just confused over the definition given in our lecturers because she wrote "the norm"

Answer 14

i don't know that book.

but if she didn't specify the inner product, then (i believe) she was being very general.