Can Ladies learn Vedas?
I am definitely not competent to answer this question. It is a generally held opinion that they are not eligible to learn Vedas. Arya Samaj a Vedic religion started by Swamy Dhayananda SAraswathi thought otherwise .He has trained large number of Women in Vedas and many of them act as priests in Arya Samaj . The very orthodox Maharashtrian community of Pune is not only teaching Vedas to them but very large number of women act as priests to that community.(refer http://www.ipsnews.net/2009/11/india-women-as-hindu-priests-have-an-edge/ )
Four Women Vedic sages do appear in several Upanishads in arguments with men sages. They are Gosha, LOpamudhra , Maithreyi and Gargi . Gargi in an open forum challenges the great sage Yagnavalkya for argument about Vedic truths. Parashara Smriti defines Brahmavadini as a lady who studies Vedas after Upanayana . There are also a class of women in the Vedas called Rishikas( Women sages). The Harita Dharmasutra (of the Maitrayaniya school of Yayurveda) declares that there are two kind of Status to Women women: Sadhyavadhu who marry, and the Brahmavaadini who are inclined to religion, they can wear the sacred thread, perform rituals like the agnihotra and read the Vedas. Bhavabhutis Uttararamacharita 2.3 says that Atreyi went to Southern India where she studied the Vedas and Indian philosophy. Shankara debated with the female philosopher Ubhaya Bharati, and Madhavas Shankaradigvijaya (9.63) mentions that she was well versed in the Vedas. Tirukkoneri Dasyai (15th 10 century) wrote a commentary on Nammalvars Tiruvaayamoli, with reference to Vedic texts like the Taittiriya Yajurveda. Any way these are subject to argument and as I told , I am not competent to tell my opinion about it
But I would like to draw your attention to the legend in Temple at Uthirakosa mangai in Ramnad about Lord Shiva teaching Vedas to goddess Parvathi. When the Lord taught the Goddess Vedas , she was not able to repeat them. So God Parameshwara cursed Goddess Parvathy to become daughter in a vedic family in Ramnad . He told her that only after she passes in the test of Vedas after she learns it she will be taken back,. Goddess as the lady Poonmulayaal learns Vedas from her scholar father and when Lord Shiva came to marry her , she was able to answer his questions in a proper manner tHe marriage takes place but when Lord Shiva further tested her knowledge of the Vedic Manthras she failed.But then the lord taught her the correct import of all Vedic Manthras. Uthira means answer and Kosa means Vedas and since the lady Answered about Vedas the place was called Uthirakosamangai. Since Lord Shiva married her there it is also called Kalyana pura.
( Note on the temple:The God in this temple is called Mangalanayakar and the Goddess Poon mulai ammai. There is a saying in tamil “Mann mundhiyo , mangai MUndhiyo” indicating that Goddess parvathy is more ancient than the mud.Another greatness of this temple is its a very huge six feet emerald Nataraja statue which is always covered with sandal paste . On Arudhra darsanam day , however you can see God Nataraja in his green glory. No musical instruments are played in the temple fearing that it may damage the emerald Nataraja.Also there is sahasralingam at this temple)