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No permission is required because the dust jacket was first published prior to 1978 without a valid copyright notice. The Door into Summer was first published in 1957; the hardcover book itself carried a copyright notice, so its contents remain copyrighted. However, the first-edition dust jacket did not carry a separate copyright notice. According to the Compendium of U.S. Copyright Office Practices: Chapter 2200, § 2207.1(C) at p. 15:

"A notice of copyright on the dust jacket of a book is not an acceptable notice for the book, because the dust jacket is not permanently attached to the book. Likewise, a notice appearing in a book is not an acceptable notice for the dust jacket or any material appearing on that dust jacket, even if the book refers to the jacket or material appearing on the jacket."

Keep in mind that the pre-1989 requirements for copyright notice were highly formalistic and, other than a few enumerated exceptions, required these three elements:

1. "The symbol © or the word 'Copyright' or the abbreviation 'Copr.' or an acceptable variant such as "(c)";
2. "The year of first publication for the work"; and
3. "The name of the copyright owner, or an abbreviation by which the name can be recognized, or a generally known alternative designation of the owner."

If just one of these elements is omitted, the work is deemed to be published without notice and is not eligible for copyright protection. A proper copyright notice is not visible anywhere on the dust jacket. Credits like "Jacket design by Mel Hunter" do not meet the requirements, nor do the identifications of the publisher and author.