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Notes :Corollary: Let $L\subset X$ be a Banach space, then there exists a complement of $L$ that is closed and can be defined by linear functionals linearly independent.

Proof: Consider $e_1,e_2...e_n$ a basis of $L$. Then $f_i(e_k)=\delta_{ik}$ delta Kronecker.

$x=x_1+x_2$ and $x_1=\sum_\limits{i=1}^{n}f_i(x_1)e_i$

Therefore

$x_2=x-x_1=x-\sum_\limits{i=1}^{n}f_i(x_1)e_i$

Consider $x_2\in M$ $0=x_1+x_2$ and $x_1\in L,\:\:x_2\in M$

$x_2=\sum_\limits{i=1}^{n}c_ie_i$

$c_k=f_k(x_2)=0$ and $x_1=x_2=0$

Then $L\oplus M$

Question:

1) I stop understanding at the point the author states $0=x_1+x_2$. How does the author goes from $0=x_1+x_2$ to $L\oplus M$? What is the logic behind $0=x_1+x_2$?

2) After proving the Hahn-Banach extension theorem. I am going through a lot corollaries about functionals like this one. However I have not found them in the literature within my grasp. Perhaps because these corollaries are called by another name. Could someone point me in the direction of finding these corollaries?

Thanks in advance!

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  • $\begingroup$ Is this the proof you've found word-for-word? $\endgroup$ – Aweygan May 5 '18 at 15:02
  • $\begingroup$ @Aweygan This is a notebook proof from a functional analysis class. I added the words to make it more understandable. Thanks for your comment! $\endgroup$ – Pedro Gomes May 5 '18 at 15:04
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    $\begingroup$ You can find a better, more complete proof of this statement here. This particular corollary is in Rudin's Functional Analysis. Note that every Banach space is a locally convex TVS. $\endgroup$ – Michael Lee May 5 '18 at 15:16
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First, I am compelled to say that these notes are lacking a lot of detail. If this is what you've copied from a proof in class, then I must recommend you write down more information when you take notes.

I believe what's happening at the point where you get confused is that we assume $x_1\in L$, $x_2\in M$, and $x_1+x_2=0$. If you can then show that $x_1=x_2=0$, then it follows that $L\cap M=\{0\}$. And assuming you've already shown that $L$ and $M$ are closed and $X=L+M$, then you have $X=L\oplus M$.

Per your second question, virtually every book on introductory functional analysis would contain the corollaries you're after. To name a few that I like, Rudin's book Functional Analysis, Conway's A Course in Functional Analysis, and Folland's Real Analysis are great resources.

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  • $\begingroup$ I used Folland for my graduate real analysis course and the first bit of functional analysis (with Brezis for the rest). Honestly, I'm not crazy about it and would probably replace it with Kreyszig, Introductory Functional Analysis with Applications. $\endgroup$ – Michael Lee May 5 '18 at 15:20
  • $\begingroup$ I'll edit my answer to include that, but I have the opposite opinion. I love Folland's book (we also used in it graduate real analysis), and dislike Kreyszig's book. $\endgroup$ – Aweygan May 5 '18 at 15:24
  • $\begingroup$ I am following Kreyzig's Book and I found nothing about these corollaries. $\endgroup$ – Pedro Gomes May 5 '18 at 15:26
  • $\begingroup$ @PedroGomes Yet another reason to dislike Kreyszig's book: it's coverage of the Hahn-Banach theorem is definitely inadequate. Go to another book I mention. Try Conway's book. $\endgroup$ – Aweygan May 5 '18 at 15:29
  • $\begingroup$ Could you please tell me $f_k(x_2)=0=c_k$ but here the author meant $f_i(x_2)=0=c_i$. I mean the functionals that acted in $L$, right? $\endgroup$ – Pedro Gomes May 5 '18 at 15:50

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