Hadith: The Prophet (may Allah's peace and blessings be upon him) as well as Abu Bakr and ‘Umar, used to begin the prayer with {All praise is due to Allah, Lord of the worlds}

عن أنس رضي الله عنه : أن النبي صلى الله عليه وسلم وأبا بكر وعمر كانوا يفتتحون الصلاة بـ﴿الحمد لله رب العالمين﴾. زاد مسلم: لا يذكرون: ﴿بسم الله الرحمن الرحيم﴾ في أول قراءة ولا في آخرها. وفي رواية لأحمد والنسائي وابن خزيمة: لا يجهرون بـ﴿بسم الله الرحمن الرحيم﴾. وفي أخرى لابن خزيمة: كانوا يُسرُّون.
[صحيح بكل رواياته] - [الرواية الأولى: متفق عليها. الرواية الثانية: "لا يجهرون" رواها أحمد والنسائي وابن خزيمة]

Anas (may Allah be pleased with him) reported that the Prophet (may Allah's peace and blessings be upon him) as well as Abu Bakr and ‘Umar, used to begin the prayer with {All praise is due to Allah, Lord of the worlds}Lord of the worlds)". Imām Muslim added: "They did not recite 'In the name of Allah, the Most Beneficent, the Most Merciful' at the beginning of the recitation or at the end of it." Another narration reported by Ahmad, An-Nasā'i, and ibn Khuzaymah reads: "They did not recite 'In the name of Allah, the Most Beneficent, the Most Merciful' loudly." Another narration reported by Ibn Khuzaymah reads: "They used to recite it silently." This explains the negation mentioned in Imām Muslim's narration and answers the questions raised by scholars about it.
[Sahih/Authentic with all its versions] - [Ibn Khuzaymah - An-Nasaa’i - Narrated by Ahmad - Narrated by Bukhari & Muslim]

Explanation

The noble Hadīth indicates that the Prophet (may Allah's peace and blessings be upon him) and his two Companions (may Allah be pleased with both them) did not use to recite "In the name of Allah, the Most Beneficent, the Most Merciful" loudly at the beginning of Sūrat al-Fātihah during the prayer. This confirms that Basmalah is not part of Sūrat al-Fātihah. The Permanent Committee for Issuing Fatwas stated: "The sound opinion is that the Basmalah is not part of Sūrat al-Fātihah nor any other Sūrah. Rather, it is an independent verse in the Qur'an, and part of a verse in Sūrat An-Naml, which reads: {Indeed, it is from Sulaymān, and it reads: {In the name of Allah, the Most Beneficent, the Most Merciful} [Sūrat An-Naml: 30]. It is recommended to recite it at the beginning of every Sūrah, except Sūrat At-Tawbah. It is an act of Sunnah to recite it silently during the prayer before Sūrat al-Fātihah."