Homework Statement
Q1)
One way to define a system of coordinants for a Sphere S^2 given by x^2 + y^2 + (z-1)^2 = 1 is socalled stereographical projection \pi \thilde \{N} \rightarrow R^2 which carries a point p=(x,y,z) of the sphere minus the Northpole (0,0,2) onto the intersection...
Homework Statement
A given sphere S^2 is given by x^2 + y^2 + (z-1)^2 = 1
where stereographical projection \pi:\pi: S^2 \thilde \{N\} \rightarrow \mathbb{R}^2
which carries a point p = (x,y,z) of the sphere minus Northpole N = (0,0,2) onto the intersection of the xy plane which a straight...
Homework Statement
Hi I have justed switched to a new subject and have some question.
1) Show that if X is a topology space then there exist an equivalence relation if and only if there exist a connected subset which contains both x and y.
2) Show that the connected components are a...
Just let's take it from the top. My understand that M can considered at a subspace of R_infinity. That hopefully correct?
So what You are simply saying is simply to show that this is infact the case then I say for any real scalar S in M then S \cdot \sum_{n=1}^\infty} r_n^2 < \infty ...
Anyway I just thought about something after re-reading my linear algebra book and talked to my professor he said something like this:
An infinite dimensional Vector Space is defined as
\mathbb{R}^\infty = \{(v_1, v_2, \ldots)| v_n \in \mathbb{R}, \forall n\}
Then if Our M is of...
My space M isn't that an Euclidian N-Space? Because that deals also with a sequence of real valued vectors. Thusly making into an Vector Space in R^n ?
The question is as follows:
Is says that Let M = \{\{r_n\}_{n=1}^{\infty}| r_n \in \mathbb{R}, \sum_{n=1}^{\infty} r_n^2 <\infty\} Show that M is a Vectorspace with the inner product
\langle \{r_n\}_{n=1}^{\infty} \{s_n\}_{n=1}^{\infty} \rangle = \sum_{n=1}^{\infty} r_n \cdot s_n
and...
Hi Mark,
Thank You for your reply. (Maybe this is a stupid question) But how can M be seen as a Vector Space if it doesn't belong to R^n?
Would you say its enough to show that the three axioms of inner product true and then conclude "Hence that the inner product between (r_n, p_n) exist...
Homework Statement
I have posted simular questions a couple of times but now I feel I have a better understanding(hopefully).
Given a Vectorspace M which is defined as a sequence of realnumber \{r_n\} and where \sum_{r=1}^{\infty} r_n < \infty
Show that M has an innerproduct given by...
I was simply trying to understand why this sum of squared elements of the series {x_j}
is put into the assigment? And why its less than infinity? I understand that maybe this square respresents that the inner product between elements in the above series.
Isn't the series the same as \langle...
Homework Statement
Given the vectorspace consisting of a realvalued sequences \{x_j\} where \sum_{j=1}^{\infty} x_j^2 < \infty . Show that M the vectorspace has an innerproduct given by
\langle \{x_j\}, \{y_j\}\rangle = \sum_{j=1}^\infty x_j \cdot y_j
Homework Equations
Since...
Homework Statement
I seem to remember that a parameterized a(t) curve in \mathbb{R}^3 that one can construct the tangent from the slope of a'(t) and the curve itself.
such that the tangent line L = a(t) + s * a'(t) to a. This is supposedly a straight line in \mathbb{R}^3.
To make a...
Question.
Does all axioms of the product have to shown in context with the desired space M in order to show that the inner product and x and y satisfies the definition?
Yes with respect to the definition that M is Vector Space consisting of real numbers with
\sum_{n = 1}^{\infty} x_n^2 < \infty. This last part that means that the squared normed space on M is less than infinity? Thus the normed space is not infinitely large and then its possible to test the...