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I've started to think about the dual $(\mathbb{R}^n)^*$ of all linear continuous functionals on $\mathbb{R}^n$ as a subspace of all real valued continous functions on $\mathbb{R}^n$. As I understand it the linear space of all real valued continuous functions from a closed interval $I\subset \mathbb{R} ^n$, denoted here by $C[I]$, is infinite dimensional for every $n$. Since, for example the functions $\{1,x, \cdots x^m\}$ in $C[I]$ are linearly independent for every $m\in \mathbb{N}$. But the space of all linear continuous real valued functions from any subset $I\subset \mathbb{R}^n$, (i.e $I^*$) is finite dimensional. With (the dual) basis $\{f_i\}_{i=1}^n$ where $f_i(e_j)=\theta_{ij}$. Where $\{e_{j}\}_{j=1}^n$ is a basis for $\mathbb{R}^n$ and $\theta_{ij}=1$ whenever $i=j$ and zero othervise. Hence the dual space $I^*$ is a very small subset of $C[I]$? How can this be understood more intuitively..? I mean how can we "explain" the difference in size, caused by the extra assumption of linearity?

harajm
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    You seem to have explained it pretty well. Is there some part of your explanation that disagrees with your intuition? – Ben Grossmann Nov 30 '13 at 00:52
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    How do you explain how there are a lot more curves than just straight lines? – tomasz Nov 30 '13 at 00:54
  • I was a bit unsure that this description of the situation was correct, so wanted to put it here to make sure. – harajm Nov 30 '13 at 00:58

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To determine a linear functional in $\mathbb R^n$ you need $n$ numbers. To determine a continuous function, you need infinitely many.

Martin Argerami
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