Last time we started a little bit discussing

Bochner spaces that is function spaces which live on a time interval but have

their values in other Banach spaces which is supposed to mimic then the

spatial dependence of time and space dependent functions and we have

introduced the LP spaces and in a second step the W1P spaces these are the

correspondence of the so to speak classical Zoppolev spaces, seen various

relationships and now what we still need is a relationship to the space of

continuous functions because we need continuity in time to express in a

strong sense what it does it mean that such a solution obeys an initial data.

We have can introduce and have introduced already a notion of a trace in time

which goes just by partial integration so we integrate so we need corresponding

test functions which allow this point in time where we want to have the trace to

be in their support and not the say the final time such that only one

boundary part which is just an evaluation here in the time for the time

interval appears so this we can also do but if we want to have a strong a strong

notion of an initial data we need continuous function so we need

something like so the general situation is X is a Banach space

if there are any doubts it is also reflexible for some results we need

for some results we need this as we have discussed so not not for everything so

we have our time interval let's say 0 t of course it need not to be this initial

point is the time interval so this is the general setting and now we look at

the space c0 on i with values in X which just means as we are used to do to have

it and so there's no new consideration concerning measurability or things like

that necessary because continuity is a notion which we can write down in

general having Banach spaces having so to speak two bars instead of one bar for

defining the distance so we have here all those functions were which are continuous

okay so the norm correspondingly is the what we would also in the in the scalar

case define as a norm and in this norm we then also here get a Banach space is in

general all those spaces which we have introduced turned out to be Banach

spaces so the norm is just take the supremum as usual but now it's not the

modulus or a vector norm it's just the Banach space norm of the images and we

go over the full here we have done the X norm and here we go over the full

integral sorry interval and we have discussed already the the fundamental

calculus of differentiation integration if you want and in the general case if a

W what turns out here is in general we have this this this situation so if we

integrate we have if we have the time derivative in some weak sense and let me

remind you the way we had it what didn't we have it already so so maybe we have

not yet touched upon this so what we now here can put down so as we have here a

con in in a general case we can only we only have this fundamental theorem for

almost all instances in time t1 and t2 and here because of the continuity we

have it in general and that means in particular an embedding theorem maybe I

write it down it's the following so the general assumptions so what do we need

we need a u from l1 0 t X we need its derivative it's if I write down the

derivative I always mean the distribution of derivative which can be

represented by a function and the function is supposed to be again an l1

function 0 t X so if you want what we have here what we have here is we have a

u from W 1 1 0 t X so we have one weak derivative and this one derivative is an

l1 so that is the most in some sense that's the most weak notion of solution

as long as we insist that and that now it becomes a little bit tricky I admit

it as long as we exist that all the image spaces are the same so our typical