Q1) Let $E$ a Banach space of infinite dimension. Could someone gives me an example of linear form that is not continuous ? I can't find any.

Q2) Let $E$ and $F$ Banach spaces. If it exist (and if it doesn't exist, could you tell me why) give an example of a function $f\in \mathcal L(E,F)$ that is not continuous if

1) $\dim(E)=\infty $ and $\dim(F)<\infty $.

2) $\dim(E)<\infty $ and $\dim(F)=\infty $.

3) $\dim(E)=\infty $ and $\dim(F)=\infty $.

I know that if $\dim(E)<\infty $ and $\dim(F)<\infty $, then such function doesn't exist.


closed as off-topic by José Carlos Santos, Tyrone, Mathmo123, Taroccoesbrocco, Rhys Steele Aug 1 '18 at 11:33

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    $\begingroup$ Did you even bother to do a search here for “discontinuous linear form”? $\endgroup$ – José Carlos Santos Jul 31 '18 at 17:58

Q1) Take a basis $\{e_i\}_{i\in I}$. By replacing $e_i$ by $e_i/\|e_i\|$ we can assume that $\|e_i\|=1$. Define $f(e_i)$ in any way you want and $f$ can be extended to a linear function. In particular, of you define $f(e_i)$ to take an unbounded set of values, then $f$ will be unbounded.

Q2) 1) If $F$ is the zero vector space, then all linear functions are zero, and therefore continuous. Otherwise, you can pick a vector $0\neq v\in F$ and define $g:E\to F$ sending $F(x)=f(x)v$, where $f$ is the linear function from (Q1).

Q2) 3) The same argument as the previous one.

Q2) 2) If $dim(E)<\infty$ then for all linear $f:E\to F$ we have that $f(E)$ is finite dimensional, and therefore closed in $F$. It follows that $f$ is continuous if and only if $\hat{f}:E\to f(F)$ defined by $\hat{f}(x)=f(x)$ is continuous. But $f$ is a linear map between finite dimensional Banach spaces. It is therefore continuous.


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