Yup.math8 said:*I can only think about the metric space ([0,1],d*) where d* is the discrete metric. [0,1] is bounded but not totally bounded.
I know [0,1] is closed but I'm not sure if ([0,1],d*) is complete. (I know a space is complete if every Cauchy sequence in there converges).
I guess since a metric space is compact iff it is complete and totally bounded., we get that [0,1] is closed and bounded, but since it is not totally bounded, it cannot be compact.
We can come up with a simple example where f_n -> 0 in L^2[0,1] but f_n doesn't converge pointwise anywhere. Here's how: let f_1 = characteristic function of [0,1/2], f_2 = characteristic function of [1/2,1], f_3 = characteristic function of [0,1/3], f_4 = characteristic function of [1/3,2/3], f_5 = characteristic function of [2/3,1], and so on. I'll let you check that this does the job.*I think (fn) converges to f in L^2([0,1]) if lim as n tends to infinity of integral of |fn-f|^2 dm is 0.
My problem is that I cannot think of a not really complicated example.
An example is a specific instance or case that illustrates a concept or idea. A counterexample is an example that proves a statement or hypothesis to be false.
Examples and counterexamples play a crucial role in science as they help to validate or disprove theories and hypotheses. They provide evidence to support or refute scientific claims, and can lead to the development of new theories or modifications of existing ones.
A strong counterexample is one that disproves a statement or hypothesis in all cases, while a weak counterexample only disproves it in some instances. In science, strong counterexamples hold more weight in challenging and modifying theories.
Examples and counterexamples can be used in scientific experiments to test hypotheses and theories. Scientists can design experiments to provide examples that support their hypotheses, and also look for counterexamples that may disprove them.
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