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Eulogy

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## Homework Statement

Let G = { f [itex]\in[/itex] C[0,1] : [itex]^{0}_{1}[/itex][itex]\int[/itex]|f(x)|dx [itex]\leq[/itex] 1 }

Endowed with the metric d(f,h) = [itex]^{0}_{1}[/itex][itex]\int[/itex]|f(x)-h(x)|dx. Is G totally bounded? Prove or provide counterexample

**2. Relevant Theorems**

Arzela-Ascoli Theorem, Theorems relating to compactness, equicontinuity etc and

Let M be a compact metric space. A subset of

C(M) is totally bounded iff it is bounded and equicontinuous. (in terms of the sup metric)

## The Attempt at a Solution

I'm not to sure whether G is totally bounded or not to begin with. If I was working in the sup metric the subset is not totally bounded as I can find a family of functions which are not equicontinuous in G. However I'm not sure how or if this translates to the integral metric defined above. Any help would be much appreciated!