Gemara

Beraisa: "Lachem" teaches that on the first day one is not Yotzei with another's Lulav, unless he gave it for a gift.

A case occurred, Chachamim were on a ship and only R. Gamliel had a Lulav, which he had bought for 1000 Zuz. He was Yotzei with it, then gave it for a gift to R. Yehoshua. R. Yehoshua was Yotzei with it and gave it to R. Elazar ben Azaryah, who was Yotzei with it and gave it to R. Akiva. R. Akiva was Yotzei with it and returned it to R. Gamliel.

Question: Why must it teach that he returned it?

Answer: This teaches that a gift on condition to return it is a valid gift.

Rava: If one said 'Take this Esrog on condition that you return it to me', and the recipient took it for the Mitzvah, he was Yotzei only if he returned it afterwards.

Bava Basra 137a - Rav Nachman bar Rav Chisda: If Reuven told Shimon 'I give to you this Esrog for a gift, and after you to Ploni', and Shimon took it to be Yotzei, Rebbi and R. Shimon ben Gamliel argue about whether or not he was Yotzei.

Objection (Rav Nachman bar Yitzchak): The Tana'im argue only about whether or not Kinyan Peros k'Kinyan ha'Guf;

Here, all agree that (Reuven is Makneh it so that) Shimon is Yotzei. If not, what was the point of the gift?!

Rishonim

The Rif and Rosh (3:30) bring the Gemara.

Rambam (Hilchos Lulav 8:10): On the first day one is not Yotzei with another's Lulav, unless he gave it for a gift.

Ba'al ha'Itur (brought in Rosh 3:30): Since the recipient knows that the giver has no other Lulav and he needs it back, there is no need to specify. Rather, he gives it Stam (without specifying) and the recipient returns it Stam, just like Chachamim on the boat.

Rosh (ibid.): Surely, a gift on condition to return is better. The Ba'al ha'Itur teaches that even if it was given Stam, it was on condition to return it.

Note: According to our text of the Rosh 'Yoser Tov Matanah Al Menas...'; the Rosh explains why the Ba'al ha'Itur teaches about giving Stam, when there is a better method. The Gilyon Maharsha's text read 'Yoser Tov mi'Matanah Al Menas...', i.e. Stam is better than a gift on condition to return, and the Rosh explains why the Ba'al ha'Itur needed to teach about giving Stam.

Question (Gilyon Maharsha YD 305 on Shach 6): Why is Stam clearly better than a gift on condition to return? It is a Chidush that we assume that the giver knows that the Halachah, that he must give for a gift (and not for a loan)!

Poskim

Shulchan Aruch (OC 649:5): After the first day one may take another's Lulav without his knowledge. A person wants others to do Mitzvos with his money, so it is like a loan.

Source (Gra DH u'Mutar): In Pesachim (4b) and Bechoros (58a) it says that a person is happy to do Mitzvos with his money.

Mishnah Berurah (32): Even though the owner is happy, this is no better than a loan, therefore it does not help on the first day.

Kaf ha'Chayim (63): We do not say that in order to enable another to do a Mitzvah, one is happy to give his money as a gift on condition to return.

Mishnah Berurah (34): This is unlike borrowing a Sefer without permission, which is forbidden (OC 14:4). One is concerned lest his Sefer get ruined. Here, there is no concern lest the Lulav get damaged while taking and shaking it.

Note: Nowadays, many permit borrowing printed Seforim without permission, because they are much cheaper than Seforim used to be and they do not wear out as easily. It would seem that the Heter to borrow a Lulav without permission is only if he will shake it gently. The Mishnah Berurah (651:43) criticizes those who shake it so vigorously that it almost breaks. One should be extra careful if the top is covered with bark, for if it falls off the Lulav might become Pasul!

Mishnah Berurah (34): One may borrow without permission once, but not every day, for then the owner might be Makpid. Similarly, the borrower should not himself give to someone else to borrow. If the owner is around one must ask him, for we do not rely on the Chazakah when it can be easily verified. Surely, if one knows that the owner would be Makpid it is forbidden. One must take the Lulav where it is. He may not take it from the house to the Beis ha'Keneses or vice-versa.

Kaf ha'Chayim (62): If Reuven forcibly took Shimon's Esrog (after the first day) to be Yotzei and return it immediately, he was Yotzei. Such a Sho'el she'Lo mi'Da'as is not considered a Gazlan, since he does not take it for Hana'as Mamon. He is not liable for Ones.

Shulchan Aruch (658:3): On the first day one is not Yotzei with a borrowed Lulav, for it is not "Lachem".

Taz (649:11): In Chutz la'Aretz, a borrowed Lulav is Pasul on the second day mi'Safek.

Magen Avraham (3): If Reuven was away and his wife gave his Lulav to his friend Shimon for a gift, Shimon was not Yotzei, for perhaps Reuven would prefer to sell the Lulav for the day. She may sell his Lulav, because Zachin l'Adam she'Lo b'Fanav. However, if Reuven is distinguished it belittles him to sell his Lulav, therefore she may not sell it but she may give it.

Rebuttal (Bechori Yakov, brought in Mishnah Berurah 8): Being pleased with the loan helps only after the first day (649:5). Even if the husband found out later and approved, Shimon was not Yotzei. This is even if his wife buys and sells for the needs of the house.

Magen Avraham (3): According to the Rosh in Kidushin (1:20), if Reuven knows that a gift is required yet he said that he lends it, Shimon is not Yotzei. According to the Rosh in Bava Basra, Shimon is Yotzei.

Note: The Gemara (30a) explicitly says that one is not Yotzei with a borrowed Lulav. The Rosh in Bava Basra must explain that this is when the lender did not know that the borrower wanted to be Yotzei with it, e.g. he lent it long before Sukos.

Question: The Shulchan Aruch (14:4) permits to borrow another's Talis without his knowledge and bless on it. The Rosh (ibid.) holds that if one borrowed a Keli to be Mekadesh a woman, she is Vadai Mekudeshes, because the lender knew that the Mekadesh must own it. Why is Lulav different?

Answer (Taz 14:5 DH veha'Nir'eh): A person is concerned lest his Lulav become Pasul, therefore his knowledge is required. When one says 'I lend to you my Lulav' we do not assume that he intends for a gift, because the Torah says "Lachem" to disqualify a borrowed Lulav. Regarding Kidushin there is no verse. Reasoning dictates that one must Mekadesh with his money, therefore we assume that the lender intended for a way that helps. However, also regarding Tzitzis it says "Kesuscha"! Perhaps one should not bless on a Talis unless it was given for a gift, even though the Rosh and Shulchan Aruch permit.

Mishnah Berurah (9): Poskim argue about this, so one should repeat the Mitzvah (with a gift on condition to return) without a Berachah.

Mishnah Berurah (10): One should be careful to pay for the four Minim before Yom Tov. If he did not, and did not take the Minim to his Reshus to acquire through Kinyan Chatzer, mid'Oraisa they are not his!

Kaf ha'Chayim (14,15): Most Acharonim say that the Shulchan Aruch holds that a Kinyan mid'Rabanan helps for mid'Oraisa laws, but the Rambam argues, so one should be stringent. One must also do Meshichah to his Reshus, for mid'Rabanan money alone does not acquire.

Kaf ha'Chayim (16): If one gives to a married woman, he should specify that it is on condition that her husband has no rights to it (otherwise, it is not fully hers).

Shulchan Aruch (ibid.): Even if the giver said 'this is a gift until you are Yotzei, and then it is mine again', the recipient is not Yotzei, for this is like a loan. If he gave it for a gift, the recipient is Yotzei.

Shulchan Aruch (5): If Reuven gave it Stam, it is as if he said on condition that you return it to me. We assume so because Reuven has no other Lulav and needs it back.

Rema: Shimon must return it for a gift, so Reuven can be Yotzei with it.

Source (Gra DH v'Tzarich): This is why one should not give his Lulav to a child.

Shulchan Aruch (ibid.): Even if Shimon gave it to Levi, and Lulav to Yehudah, as long as Reuven gets it back in the end Shimon was Yotzei.

Magen Avraham (7): This is even if Reuven said 'on condition to return it