Talk:Dual space of C 0(x) vs C b(x)

Beginning

• The phrase at the beginning is not a complete sentence.

Background and Statement

• Comment that, if X is compact, all continuous functions belong to C0.
• ``In other words, this is the space`` should be a new sentece.
• ``... and let Cb(X) be the `` should be a new sentence
• It seems a little unbalanced that you specify the norm for Cb but not C0.
• Cite the Rudin reference using wikimedia's ref syntax
• There are several periods missing in the statement of Rudin's theorem.
• Specify what is meant by the notation ' (i.e. the dual space, all bounded linear functions).
• Switch the order of the last two sentences in this section.

The case of Cb(X)

• It seems a little unbalanced that the dual space of C0 is discussed in the previous section and the dual space of Cb is discussed in this section.
• The first sentence is too similar to Santambrogio and could be considered copyright infringement. It also isn't well motivated. Why does one want to produce such functions? Cite Santambrogio.
• You should cite the precise pages in Villani's book that you reference.
• The second to last sentence is not a run on sentence.
• You should change the phrase ``Hahn-Banach`` to link to the Wikipedia page on Hahn Banach. You should elaborate on why one can build a continuous extension to all f Cb.
• The last sentence is great!

Kantorovich Duality for Cb

• Cite pages of Villani, using the ref syntax
• Also create references to the Kantorovich duality pages on the OT wiki
• This section is too similar to Villani's book and should be either replaced with something else or rewritten. Perhaps it would be better to elaborate on the interesting example you began discussing in the previous section on ``The case of Cb``?