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aurorasky

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In summary, it is true that any linear map between two arbitrary finite-dimensional vector spaces is continuous. Differentiability can be determined by finding a linear map that approximates the map well.

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aurorasky

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Hi aurorasky!

This is indeed true! Take an arbitrary linear map [itex]f:\mathbb{R}^n\rightarrow \rightarrow{R}^m[/itex]. Then it suffices that the components f_{1},...,f_{m} are continuous. And this is true since

[tex]\begin{eqnarray*}

|f_1(x_1,...,x_n)|

& = & |x_1f_i(1,0,...,0)+...+x_nf_i(0,...,0,1))|\\

& \leq & |x_1||f(1,0,...,0)|+...+|x_n||f(0,...,0,1)|\\

& \leq & \|(x_1,...,x_n)\|_\infty(|f(1,0,...,0)|+...+|f(0,...,0,1)|)\\

\end{eqnarray*}[/tex]

which implies continuity. For differentiability, we have to find a linear map u such that

[tex]\lim_{h\rightarrow 0}{\frac{f(x_0+h)-f(x_0)-u(h)}{\|h\|}}=0[/tex]

take u=f and it's easy to see that this is true.

aurorasky said:

This is indeed true! Take an arbitrary linear map [itex]f:\mathbb{R}^n\rightarrow \rightarrow{R}^m[/itex]. Then it suffices that the components f

[tex]\begin{eqnarray*}

|f_1(x_1,...,x_n)|

& = & |x_1f_i(1,0,...,0)+...+x_nf_i(0,...,0,1))|\\

& \leq & |x_1||f(1,0,...,0)|+...+|x_n||f(0,...,0,1)|\\

& \leq & \|(x_1,...,x_n)\|_\infty(|f(1,0,...,0)|+...+|f(0,...,0,1)|)\\

\end{eqnarray*}[/tex]

which implies continuity. For differentiability, we have to find a linear map u such that

[tex]\lim_{h\rightarrow 0}{\frac{f(x_0+h)-f(x_0)-u(h)}{\|h\|}}=0[/tex]

take u=f and it's easy to see that this is true.

Last edited:

- #3

aurorasky

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BTW, could you explain to me how to type latex code?

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aurorasky said:Thanks! Also, if a map is differentiable, it is also continuous and so the proof can be made really simple, right?

Yes, but the proof that a differentiable map is continuous uses that linear maps are continuous... (at least the proof I've seen)

BTW, could you explain to me how to type latex code?

Click on the LaTeX symbols to see what I did.

- #5

mathwonk

Science Advisor

Homework Helper

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then you have to give the differentiable structure, which again is rpesumably the one defined by a linear isomorphism with R^n. Now of coiurse for it to be meaningful, you need to know that every linear isomorphism defines the same differentiable structure, which is equivalent to your original question for R^n, i,.e is every linear map on R^n differentible,

well by definition a map if differentiable if locally it is approximated very well by a linear map, so duhh,... yes a linear map is differentiable.

more precisely a map f is tangent to zero if |f(x)|/|x|-->0 as x-->0. and a map g is differentiable at a if there is a linear map L such that the map g(a+x)-g(a)-L(x) is tangent to zero. Obviously if g is linear and we take L = g, then

g(a+x)-g(a)-L(x) = 0, hence is certainly tangent to zero.

in infinite dimensions some linear maps are not (norm) continuous, and differentiability may be defined as "approximable by a continuous linear map". is this your confusion? i.e., is some nincompoop teaching you in finite dimensional differential calculus before finite dimensional calculus? (that happened to me.) (I was the nincompoop.)

A linear map is a mathematical function that preserves the structure of vector spaces. In other words, it takes in vectors as inputs and produces vectors as outputs, while also satisfying the properties of linearity.

Finite dimensional spaces are vector spaces that have a finite basis. In other words, they have a finite number of linearly independent vectors that can span the entire space.

Linear maps between finite dimensional spaces take vectors from one finite dimensional space and map them to vectors in another finite dimensional space. This is done by applying a linear transformation to the input vectors, resulting in an output vector in the other space.

Some common examples of linear maps between finite dimensional spaces include rotations, reflections, and dilations in Euclidean spaces. In linear algebra, linear transformations such as matrix multiplication and differentiation are also considered linear maps.

Linear maps between finite dimensional spaces have many applications in mathematics and science. They are essential in solving systems of linear equations, representing geometric transformations, and understanding the behavior of complex systems. They also play a crucial role in fields such as physics, engineering, and computer graphics.

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