Suppose \(\displaystyle (X,\|.\|)\) is a normed vector space. Let \(\displaystyle (X,\mathcal{T})\) be the initial topology with respect to \(\displaystyle \|.\|: X \rightarrow [0,\infty)\). Then the topology induced by the norm is not equal to \(\displaystyle \mathcal{T}\).

I've been thinking about this for some time now but I'm having real trouble proving it. Any help would be appreciated.

Best Regards,

Stiwan