@Koro But $c$ belongs to $\ell_\infty$. You're trying to show that for every $c\in\ell_\infty$ you get a linear continuous functional (which has the same norm as $c$.) In any case, I would suggest to continue in another chatroom so that it is not mixed with the other conversations here. (But not today - it is already late in my timezone.)

I posted slightly more rambly about this in this chat room -> chat.stackexchange.com/rooms/19167/modern-abstract-analysis which we can use if anyone wants to discuss further, no one else seems to care that the room exists

It seems that recently the activity in functional analysis chatroom is bigger than before. And there is no shortage of users who ask questions there. It would be nice to get more users with good knowledge of functional analysis who could help with the question which remained unanswered.

I have added a few links about inclusions between L_p and L_q in the functional analysis chat room.

@mathiu_lady It seems that this was asked on the main site. (At least if I correctly understood what you're looking for.) I have posted a link in the functional analysis chat room.

I am not sure I follow the last comment. Anyway, there is a post on the main with some counterexamples, see the links I posted here.

BTW if you have time, feel free to stop by in functional analysis chat room sometimes.

If you look at the starboard or bookmarked conversations, you can see that most of the discussions were about rather elementary stuff.

@AlessandroCodenotti Functional analysis chat room does not have too much activity. So it will be nice if you occasionally stop by. (But it seems that there are more people who are asking questions in that room. than people who are able to answer them.

@Daminark Incidentally, uncountability of Hamel basis in Banach spaces was a topic recently in functional analysis chat room. And, as expected, there is a post on main about this: Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable..

Sorry for advertising the room again - if it becomes too much, simply tell me off and I'll try to do this less frequently...

Functional analysis room was unfrozen not so long ago and it had some activity recently. If there are some people interested in this area of mathematics, it will be nice if you have a look at the room occasionally.

In case there are some people who would be interested, functional analysis chatroom got a fresh restart.