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This is an audio version of the Wikipedia Article:\nhttps://en.wikipedia.org/wiki/Closed_range_theorem\n\n\n00:00:14 1 History
00:00:32 2 Statement
00:04:03 3 Corollaries
00:05:12 4 See also
\n\n\nListening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.\n\nLearning by listening is a great way to:\n- increases imagination and understanding\n- improves your listening skills\n- improves your own spoken accent\n- learn while on the move\n- reduce eye strain\n\nNow learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.\n\nListen on Google Assistant through Extra Audio:\nhttps://assistant.google.com/services/invoke/uid/0000001a130b3f91\nOther Wikipedia audio articles at:\nhttps://www.youtube.com/results?search_query=wikipedia+tts\nUpload your own Wikipedia articles through:\nhttps://github.com/nodef/wikipedia-tts\nSpeaking Rate: 0.8279758296262053\nVoice name: en-GB-Wavenet-D\n\n\n"I cannot teach anybody anything, I can only make them think."\n- Socrates\n\n\nSUMMARY\n=======\nIn the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range.