Recent content by Cauchy1789

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...

Hi Mark what I don't get here. You say M is not a Vector Space? How is that possible since its the premise for the whole assignment?

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...

I get it now but anyway. With the above then if I need to show that M has an inner product defined by \langle{\{x_n\}, \{y_n\}}\rangle = \sum_{n = 1}^{\infty} x_n y_n So what I need to do here isn't to show since the Vector Space M is defined as \sum_{n = 1}^{\infty} x_n^2 < \infty...

What I would simply like to understand is why a sequence of real numbers defined as in my original post can be less than infinity? If M is a vectorspace is the "less than infinity" part because a Vectorspace is closed set?

Then my above series can be written as \sum_{n = 1}^{\infty} x_{n}^2 = \lim_{n \to \infty} \int_{1}^{n} x_{n} ^2 dx < \infty