Prove Finite Dimensional Normed Vector Space is Differentiable

In summary, the conversation discusses the differentiability of the function f: U-->U defined by f(T)= T-1, where U is the set of invertible elements in a finite dimensional normed vector space V. The proposition 1, which states that if ||T||<1, then the sequence Sn converges to (I-T)-1, is used to prove that f is differentiable at each T in U. The second proposition, which states that if ||T-S||<||T-1||-1, then S is invertible, is also mentioned in the conversation. The solution to proving the differentiability of f involves first differentiating at I and then using a technique to transfer the computation to any T
  • #1
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Homework Statement



Let V be a finite dimensional normed vector space and let U= L(V)*, the set of invertible elements in L(V). Show, f:U-->U defined by f(T)= T-1 is differentiable at each T in U and moreover,
Df(T)H = -T-1HT-1
where Df(T)= f'(T).


Homework Equations



Apparently these propositions are supposed to help (which I've already proved and can use):

1.) Let V be a normed vector space and suppose T is an element of L(V). If ||T||<1, then the sequence Sn=sum(from j=0 to n) of Tj converges. Moreover, I-T is invertible, (Sn) converges to (I-T)-1, and ||(I-T)-1||<= 1/(1-||T||).

2.) Suppose V is a normed vector space and T, an element of L(V), is invertible. If S is an element of L(V) and ||T-S||<||T-1||-1, then S is invertible.


The Attempt at a Solution



I've already proven that the function q:V-->V where q(T)=T2 is differentiable at each T and Dq(T)H= TH+HT, and I tried to prove this one using the same technique, but it's getting me nowhere.
 
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  • #2
First try differentiating [tex]f[/tex] at [tex]I[/tex]; your proposition 1 should help with this. Then see if you can find a way to transfer that computation to any [tex]T \in U[/tex].
 
  • #3
Nevermind, I think I figured it out.
 

1. What is a finite dimensional normed vector space?

A finite dimensional normed vector space is a mathematical construct that consists of a finite number of vectors, where each vector has a corresponding norm (or magnitude) and satisfies certain properties, such as the triangle inequality. It is often used to model physical systems in mathematics and physics.

2. What does it mean for a normed vector space to be differentiable?

A normed vector space is differentiable if it has a well-defined derivative at every point in the space. This means that the space has a smooth and continuous structure, and it allows for the calculation of rates of change and gradients at any point in the space.

3. How do you prove that a finite dimensional normed vector space is differentiable?

To prove that a finite dimensional normed vector space is differentiable, you must show that the derivative exists at every point in the space. This can be done by showing that the limit of the difference quotient (the difference between two points in the space divided by the distance between them) exists and is unique for every pair of points in the space. This is known as the definition of differentiability.

4. What are some applications of proving that a finite dimensional normed vector space is differentiable?

Proving that a finite dimensional normed vector space is differentiable has many practical applications in mathematics and physics. It allows for the calculation of rates of change and gradients in physical systems, and it is also used in optimization problems, such as finding the shortest path between two points in a space.

5. Are there any limitations to proving that a finite dimensional normed vector space is differentiable?

One limitation of proving that a finite dimensional normed vector space is differentiable is that it only applies to spaces with a finite number of dimensions. In higher dimensional spaces, differentiability can become more complex and may require different mathematical techniques. Additionally, the proof may be more difficult or even impossible if the space has unusual or non-standard properties.

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