# Norm on a scalar

Very basic question, but one I would like a formal understanding as to why.

For a normed linear space $$(X,|| \cdot ||)$$, where $$X$$ is over the field $$\rm l\!F$$ and $$||\cdot ||:X\rightarrow\rm l\!F$$.

Want I want to understand formally is why $$||a||=|a|$$, for $$a\in \rm l\!F$$. Is this wrong instead in general?

• This is just the norm of $a$ when you think of $a$ as a vector in the one dimensional vector space $F$ over $F$. Commented Oct 19, 2022 at 21:10
• The only meaning I can give to that is: any absolute value $|~|:\mathbb F\to\mathbb R$ may be considered as a norm on the $\mathbb F$-vector space $\mathbb F.$ Commented Oct 19, 2022 at 21:14
• I agree now after reading both comments. Norm of a only has meaning if a itself is an element in X. So it's false in general and only true in very specific occasions. Commented Oct 19, 2022 at 21:29

As the comments brought to my attention, it is not true in general that the norm of a scalar is well defined. Strictly speaking, we need $$\mathbb{F}\subset X$$ for the norm of a scalar to make sense.
But there are also some cases, where one can see the scalars as elements of $$X$$ that act in the same way for scalar multiplication. For example, $$C(\mathbb{R})$$ the set for continuous real valued functions. $$(C(\mathbb{R}),||\cdot||_{max})$$ is a normed linear space and in this case one can relate the scalars to constant functions. Still, $$||a||$$ is undefined for $$a\in\mathbb{R}$$, but for a$$(x)=a$$ we have $$||$$a$$||=|a|$$.