An informal, "physicist's", explanation would go like this. If g*u=g for all g then . But you would not expect the value of g at x to depend on the values of g at points other than x itself. Therefore the function u must be zero everywhere except at 0, where it has to be so infinitely huge that the value of the integral is g(x). In other words, u has to be the Dirac delta function. But this is not an element of (and in fact is not a function at all).

If you want a more mathematical argument, then you could try something like this. let and define f by Then . When |x|>ε that gives . You can probably believe (and you may even be able to prove) that this implies that u(t)=0 whenever |x|>ε. Since ε is arbitrary, it follows that u is zero everywhere except at 0.

However, does have what is called anapproximate identity. That is a sequence of elements of norm 1 such that (in the -norm) as k→∞, for all . For example, when n=1 you can take . This gives a sequence of functions that have increasingly large sharp spikes at the origin and are very small elsewhere.