More detailed remarks :

## Question 1 :

+ A lot wrote the rank-nullity theorem for (b), but the space might not be finite-dimensional.


## Question 4 :

+ v1-v2 \in W makes sense, but v_1 = v_2 + W does not, both sides of the equality live in different spaces

+ The infimum of the distance is  attained but not necessarily unique in a Banach space (see exercise 5). It is when the space is reflexive, but this is not assumed.

+ Need to use the assumption of W being closed (in particular when showing the norm is positive definite)

+ Pythagoras is NOT ||x+y|| = ||x||+||y||, but ||x+y||^2 = ||x||^2+||y||^2.


##Question 5 :

+ There were pretty much no mistake, but I just want to point our that the interesting this in the first example is that EVERY point of c0 attains the inf.


## Question 6 :

+ Most counter-examples for f. were still open on their image.

## Question 7 :

+ A few forgot in the first part to check that when taking finite intersections, we get a product with only finitely many proper opens. A product of open sets is not enough to be in our base.

+ To prove that it is an open mapping, many only checked it on the basis, which is fine but many times it was assumed that an arbitrary element of the topology  is a product of open sets.

+ Very few people used the fact that norms are equivalent on R^n, and then used the sup norm, the open balls here are squares, which are obviously open in the product topology.