Discontinuous linear mapping between infinite-dimension vector space

[yifli]

It is known that any linear mapping between two finite dimensional normed vector space is continuous (bounded).

Can anyone give me an example of a linear mapping between two infinite dimensional normed vector space that is discontinuous?

Thanks

 

[micromass]

Hi yifli!

It is known that any linear mapping between two finite dimensional normed vector space is continuous (bounded).

Can anyone give me an example of a linear mapping between two infinite dimensional normed vector space that is discontinuous?

Thanks

The standard example is a very familiar linear mapping: differentiation. Let $X$ be the set of all real polynomials on [0,1]. Equip this with the sup-norm, i.e.

$$\|f\|_\infty=\sup_{t\in [0,1]}{|f(t)|}$$

Let

$$T:X\rightarrow X:f\rightarrow f^\prime$$

Let $p_n(x)=x^n$, then $\|p_n\|_\infty=1$, but

$\|T(p_n)\|_\infty=n\|p_n\|_\infty$

thus the operator T is not bounded.

This is a very tragic result and has a lot of bad consequences...