Functional Analysis doubt

[rakehsoran]

Hi
some one please help me with the following problem

Suppose that T_0 is the interior of a triangle in R^2 with vertices A,B,C. If T_1 is the interior of the trianlge whose vertices are midpoints of the sides of T_0, T_2 the intrior of the triangle whose vertices are midpoints of sides of T_1 and so on then what is in the set intersection(from j=0 to inf) T_j?

Thanks
Rakesh

 

[HallsofIvy]

It will be, of course, a single point. I think the centroid of the triangle.

 

[micromass]

What exactly does this have to do with functional analysis?

 

[mathwonk]

indeed it is the centroid. consider two medians of the triangle. they meet somewhere inside the triangle T0. in fact the same two medians are also medians of the triangle T1, so thy also meet inside that triangle, etc...thus the two medians meet at the unique point common to all these triangles. the same statement holds for any two medians. hence all three medians meet at the common point of all the triangles, which is therefore the centroid.

 

[blue_raver22]

Hello Rakesh,

Thank you for reaching out for assistance with your problem. This question involves the concept of functional analysis, which is a branch of mathematics that deals with the study of vector spaces and linear transformations between them. In this case, we are looking at a specific transformation that involves the interior of a triangle and its midpoints.

To answer your question, we need to understand the process of creating T_1, T_2, and so on. Each time we take the midpoints of the sides of a triangle, we create a smaller triangle within the original triangle. This process can continue infinitely, creating smaller and smaller triangles within the previous ones. Therefore, the set intersection of all these triangles (from j=0 to infinity) will be the point at which all these triangles converge, which is known as the centroid of the original triangle.

In summary, the set intersection of all the T_j triangles will be the centroid of the original triangle T_0. I hope this helps to clarify your doubt. If you have any further questions, please don't hesitate to ask. Good luck with your studies!

Best regards,