Proof that a space of all continous functions f: R\to R has no norm. We do not put any other structure to this space (like for example topology). We only treat it as a vector space. Thak You very much in advance for any help.
Proof that a space of all continous functions f: R\to R has no norm. We do not put any other structure to this space (like for example topology). We only treat it as a vector space. Thak You very much in advance for any help.
Every vector space over the scalar field $\displaystyle \mathbb R$ has a norm. Given a vector space V, let $\displaystyle \{e_\alpha\}_{\alpha\in A}$ be a Hamel basis for V. (A Hamel basis is a maximal linearly independent set. The existence of such a set will usually require the Axiom of Choice.) Then every element of V has a unique expression as a finite linear combination of basis elements.
So, for each x in V there is a unique expression $\displaystyle x = \textstyle\sum_{\alpha\in A}\lambda_\alpha e_\alpha$, where only finitely many of the coefficients $\displaystyle \lambda_\alpha$ are nonzero. Define $\displaystyle \|x\| = \textstyle\sum_{\alpha\in A}|\lambda_\alpha|.$ That defines a norm on V.
That construction works in particular when V is the space of all continuous functions on $\displaystyle \mathbb R$.