{"id":194,"date":"2026-04-19T06:06:38","date_gmt":"2026-04-19T06:06:38","guid":{"rendered":"https:\/\/borovik.net\/selecta\/?p=194"},"modified":"2026-04-19T08:29:52","modified_gmt":"2026-04-19T08:29:52","slug":"how-do-you-intuitively-explain-the-fact-that-the-bidual-of-an-infinite-dimensional-vector-space-is-bigger-than-the-vector-space-itself","status":"publish","type":"post","link":"https:\/\/borovik.net\/selecta\/2026\/04\/19\/how-do-you-intuitively-explain-the-fact-that-the-bidual-of-an-infinite-dimensional-vector-space-is-bigger-than-the-vector-space-itself\/","title":{"rendered":"How do you intuitively explain the fact that the bidual of an infinite dimensional vector space is bigger than the vector space itself?"},"content":{"rendered":"
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My answer to a question on Quora:<\/div>\n
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“How do you intuitively explain the fact that the bidual of an infinite dimensional vector space is bigger than the vector space itself?”.<\/strong><\/div>\n<\/blockquote>\n
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MY ANSWER:<\/div>\n
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It is a very deep question indeed. I have to admit that I cannot give any intuitive explanation which is simpler than a reduction to some basic set theory.<\/div>\n
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To make the question as close to the set theory as possible, let us restrict our attention to the case of a vector space V over the field F_2 of two elements. If my memory does not betray me, it is an old result by Paul Eklof that existence of a basis in an arbitrary vector space over F_2 is equivalent to the Axiom of Choice (it is easy in one direction: the Axiom of Choice, in the form of the Zorn Lemma, implies the existence of a basis). So, let us accept it, and let B be a basis in V, which means that every vector v in V is defined by its support in B, that is, by the (finite) set of elements in B which sum up to v. Therefore V has the same cardinality as the set of finite subsets of B; if B is infinite, than it is easy to prove that the set of all finite subsets of B has the same cardinality as\u00a0B, hence V has the same cardinality as B.<\/div>\n
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Now let us look at the dual space V*, that is, the set of all linear functionals from V to F_2. Each such functional is uniquely determined by its support in B, that is, by the set of basis vectors where it takes value 1. Therefore V* is in one-to-one correspondence with the set 2^B of all subsets in B. But it is a classical result by Cantor that 2^B has larger cardinality than B and hence V* has larger cardinality than V. Of course, the cardinality of the bidual V** is even larger.<\/div>\n
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The case of an arbitrary field F can be handled in a similar way, but we will need to deal with the cardinality of the set F^B.<\/div>\n
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Is this intuitive? Well, it is intuitive for me because it was the first thought that crossed my mind. But I am not a set theorists and I have no idea whether the same can be proven without the Axiom of Choice. Also, the term \u201cinfinite dimensional vector space is ambigous: does it mean \u201ca vector space without a finite basis\u201d or \u201ca vector space with an infinite basis\u201d? And can anything that intimately involves the Axiom of Choice be called intuitive?<\/div>\n
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COMMENTS: Someone commented on that: “Without Axiom of Choice, a vector space can have trivial dual (containing only the null functional).” If true (I am not a set theorist and cannot judge), this even more emphasises the crucial role of Axiom of Choice. Also, I had perhaps mention in my answer that the situation could be very different if we consider topological vector spaces and continuous linear functionals on them.<\/div>\n<\/blockquote>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

My answer to a question on Quora: “How do you intuitively explain the fact that the bidual of an infinite dimensional vector space is bigger than the vector space itself?”. MY ANSWER: It is a very deep question indeed. I have to admit that I cannot give any intuitive explanation which is simpler than a […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-194","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/194","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/comments?post=194"}],"version-history":[{"count":2,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/194\/revisions"}],"predecessor-version":[{"id":196,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/194\/revisions\/196"}],"wp:attachment":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/media?parent=194"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/categories?post=194"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/tags?post=194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}