Are girls allowed to learn the Vedas?
Yes, females are allowed to read the Vedas. I am posting a passage from a famous 14th century text(Jivanmuktiviveka of Vidyaranya) which shows the importance of women Rishis in the formulation of the present day Hinduism and the equal rights enjoyed by women in Hinduism..
Renunciation is mentioned in the Taittiriya and other Upanishads: 'Some have attained immortality, not by acts, nor by offspring, nor by wealth, but by renunciation alone' (Kaivalya Upanishad., 3). Females also are entitled to this kind of renunciation. The Caturdharike of the Mokshadharma, by using the word bhikshuki (female mendicant) with reference to the lady in question, when dealing with the controversy between Sulabha and janaka, indicated that females may renounce before marriage, or after the death of their husband and may go about as religious mendicants, may learn and hear the sastras dealing with moksha (namely the Upanishad and cognate literature), may meditate upon the atman in seclusion and assume the emblems of tri-danda, etc. In the wake of the argument in the Devatadhikarana (the section about devata) in the fourth chapter of the third book of the Sarirakabhasya, wherein a discussion has been started relating to the rights of a widower (to such renunciation), the name of the lady Vacaknavi has also been mentioned. These references go to strengthen the right claimed by Maitreyi, the wife of Yajnavalkya, of whom she inquires in the words: 'Of what use would such (wealth) be to me, if the possession of such would not render me immortal? Tell me whatever your worship knows,as to what I should do, in order to be free from mortality' (Brihadaranyaka Upanishad 4.5.4).
Jivanmuktiviveka of Vidyaranya translated by S. Subrahmanya Sastri and T.R. Srinivasa Ayangar, Chapter 1, Vividisa-samnyasa: The Renunciation of the Seeker
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