Dual Space of Real Numbers

Question

I would have a very naive question. Let $X = \mathbb R$, and define $L(X, \mathbb R):= \{g: X \rightarrow \mathbb R, g \ \text{is linear and continuous}\}$. Is it correct to say that $L$ consists in this case of all functions $g:X\rightarrow \mathbb R$ of the form $x\mapsto m\cdot x$, where $m$ is a constant? Or are there more functions than these that belong to $L(X, \mathbb R)$?

Answer 1

If $f\colon\mathbb{R}\to\mathbb{R}$ is $\mathbb{R}$-linear, set $m=f(1)$. Then, if $x\in\mathbb{R}$, by linearity you have $$ f(x)=f(x1)=xf(1)=mx $$ Conversely every such map is $\mathbb{R}$-linear (and also continuous).