Recent content by Damidami

Let's change the last statement to this one (a,b) \in A \times B would simply mean a function \phi : \{1,2\} \to C_1 \cup C_2 such that \phi(1) \in C_1 and \phi(2) \in C_2 . Wait, that is the definition of the cartesian product, isn't it? just all the functions like \phi? It that is...

Hi Stephen, Thanks for your response. Usually an indexed family of elements of X is just a function I \to X. In my example I would take X = C_1 \cup C_2. And let's change the word "pick" by the word "choose". So I mean that if you choose (a,b) \in A \times B, that means you have...

Hi all. I'm having trouble understanding the cartesian product of a (possible infinite) family of sets. Lets say \mathcal{F} = \{A_i\}_{i \in I} is a family of sets. According to wikipedia, the cartesian product of this family is the set \prod_{i \in I} A_i = \{ f : I \to \bigcup_{i...

Hi Tenshou, I agree, but my question was pointing more to if there was or not any reason why the word spectrum or spectra to be used for these two apparently completely different things: the spectra of a linear operator, and the spectra obtained by a Fourier transformation. The use of the...

Hello, Something I have some time wondering and still couldn't find the answer is to this question: if there is some relation between the Spectrum (functional analysis) and the Frequency spectrum in Fourier Analysis. Now that I think about it there seems to be a casuality the use of the...

Hi, I understand the concept of the differential of a (differentiable) function at a point as a linear transformation that "best" approximates the increment of the function there. So for example the differential of a function f : D \subseteq \mathbb{R}^2 \to \mathbb{R} could maybe be df = 8...

Hi Vargo, Thanks for your reply. I think I can see your point. By the subset of G which leaves the equivalence classes invariant, I think you mean the maximal one with that property (as the trivial susbset of G obviously leaves the classes invariant) Anyway it's interesting that any...

Given a group G acting on a set X we get an equivalence relation R on X by xRy iff x is in the orbit of y. My question is, does some form of "reciprocal" always work in the following sense: given a set X with an equivalence relation R defined on it, does it always exist some group G with some...

Finally, the whole point of this thread was to know what exactly is the structure that defines euclidean space. So is it safe to say that euclidean space is a real finite dimensional vector space oriented and with an inner product? If that is it, the only bit I still don't understand is...

But in euclidean space you don't have an exterior nor wedge product, so there has to be another way.

Hi AntsyPants and micromass, That's interesting. Even if when we change the inner product the distance between two elements change, we can still thought it as the same space because they are isometric? So in the definition of euclidean space it doesn't matter what the inner product is, but...

Hi AntsyPants, Thanks for your answer. It's the one which came closer to what I was asking. My question is, how do you calculate the area of a paralelogram spanned by two vectors in euclidean space \mathbb{R}^3 ? You should only use vector space operations and inner product operation...

Hi DonAntonio, Thanks for your answer. I'm aware of those things, but it's different from what we are discussing here. In what we call a 3-field, the three binary operations have to be defined on the same set F, we use no external 'scalar' set, so that's a different issue.

Hi micromass, Do you mean the distance between two vectors (x_1, y_1) and (x_2, y_2) ? So you are talking about \mathbb{R}^2 ? Is there some other natural thing I should know about euclidean space \mathbb{R}^2 ? I mean, that is precisely my question, what operations (structure) is...

Hi micromass, I don't understand, you mean euclidean space is simply by definition the set \mathbb{R}^n? Take for example n=2 (we have euclidean plane \mathbb{R}^2), what is then the distance between (1,1) and (2,2)? (Remember there is no notion of distance in a "naive" set) Thanks, Damián.