PROBLEM 2: In Part (a), one of the inclusions is easy.
For the reverse inclusion, use a separation argument.

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PROBLEM 3: The space should be L^1 on an interval (a,b)
with Lebesgue measure (not the sequence space ell^1).

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PROBLEM 4: Your argument can be short (2 sentences)

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PROBLEM 5: In Part (c),  the intersection of K with
the line through e and x is closed interval. Choose
y to be the endpoint that is not e. (In general,
y is not an extreme point, but it will lie on
the boundary of K. Hence it lies on a lower-dimensional 
face of K.)