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Math Help - space of all continous real functions and a norm

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    space of all continous real functions and a norm

    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.
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    Quote Originally Posted by mmm849 View Post
    Proof that a space of all continuous 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. Thank You very much in advance for any help.
    Every vector space over the scalar field \mathbb R has a norm. Given a vector space V, let \{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 x = \textstyle\sum_{\alpha\in A}\lambda_\alpha e_\alpha, where only finitely many of the coefficients \lambda_\alpha are nonzero. Define \|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 \mathbb R.
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