Hilbert And Banach dual spaces Are, and in general, all Hilbert spaces identified to their dual ?, i.e $H'\equiv H$
What about Banach spaces? And can someone please state the weak convergence in each?
A vector space (a fortiori, Hilbert space, Banach space) is isomorphic to its dual if, and only if, it is finite dimensional. For any infinite dimensional vector space, the dual space is always &quot;larger&quot; than the predual, in the precise technical sense that there exists an injection from the predual into the dual, but never a bijection.
Additionally, the isomorphism between a finite dimensional vector space and its dual is dependent on a choice of vector basis. It is not a &quot;natural&quot; isomorphism, in the language of category theory. However, the isomorphism between a vector space and its double dual (the dual of the dual) IS natural, in that it can be defined without a choice of basis.
Studying this phenomenon is anecdotally what pushed Saunders MacLane and Samuel Eilenberg to invent category theory, by the way.
Edit: I was mistaken in the above, due to an ambiguity on the notion of &quot;dimension&quot; as used by an algebraist, and as used by an analyst; and have provided in the comments below a document which clarifies my point, and explains my mistake in your specific context.
A vector space (a fortiori, Hilbert space, Banach space) is isomorphic to its dual if, and only if, it is finite dimensional. For any infinite dimensional vector space, the dual space is always &quot;larger&quot; than the predual, in the precise technical sense that there exists an injection from the predual into the dual, but never a bijection.
Additionally, the isomorphism between a finite dimensional vector space and its dual is dependent on a choice of vector basis. It is not a &quot;natural&quot; isomorphism, in the language of category theory. However, the isomorphism between a vector space and its double dual (the dual of the dual) IS natural, in that it can be defined without a choice of basis.
Studying this phenomenon is anecdotally what pushed Saunders MacLane and Samuel Eilenberg to invent category theory, by the way.
Edit: I was mistaken in the above, due to an ambiguity on the notion of &quot;dimension&quot; as used by an algebraist, and as used by an analyst; and have provided in the comments below a document which clarifies my point, and explains my mistake in your specific context.