# Thread: Infinite Dimensional Vector Space.

1. ## Infinite Dimensional Vector Space.

There is a 3 part question that I have been working on.

The first 2 parts are show that for a finite dimensional vector space:

$\displaystyle S \circ T$ is invertible if and only if $\displaystyle S$ and $\displaystyle T$ are invertible.
$\displaystyle S \circ T = I$ if and only if $\displaystyle T \circ S = I$

Theese two I have successfully proven. It is the last part I am having trouble with which is:

Give an example for each that shows that the statements are false for infinite dimensional vector spaces.

Any help would be appreciated, Thanks!

2. Hint : consider the vector space $\displaystyle V$ of sequences of real numbers with the map $\displaystyle S: (a_1,\dots,a_n,\dots) \mapsto (0,a_1,\dots,a_n,\dots)$. Show that $\displaystyle S$ has a left inverse which is not a right inverse.

3. Thanks for the hint!

Here is what I got

Let
and $\displaystyle T: (b_1,...,b_n,...) \mapsto (b_2,...,b_{n+1},...)$

$\displaystyle S \circ T$ will give us the original sequence however
$\displaystyle T \circ S$ wont. This proves the second
" if and only if " does not hold for infinite dimensional matrices.

However Im still not sure how to prove the first part " is invertible if and only if and are invertible." does not hold for infinite dimensional matrices.

4. Well you have to be careful; $\displaystyle T\circ S$ is the identity map, not $\displaystyle S\circ T$. You are right that it proves the second part of the problem.

For the first part, notice that $\displaystyle T\circ S$ is invertible (it's the identity!) but that $\displaystyle T$ is not. So the part of the statement which says "If $\displaystyle T\circ S$ is invertible, then both $\displaystyle T$ and $\displaystyle S$ are invertible" is false. The part which says "If both $\displaystyle T$ and $\displaystyle S$ are invertible, then $\displaystyle T\circ S$ is invertible" is always true; the inverse of $\displaystyle T\circ S$ is then given by $\displaystyle S^{-1}\circ T^{-1}$.