TITLE: Is there a Banach space version of Riesz representation theorem?
QUESTION [6 upvotes]: Let $X,Y$ be Banach spaces over $\mathbb{R}$ and $B:X\times Y\rightarrow \mathbb{R}$ be a continuous bilinear function such that $X\rightarrow L(Y,\mathbb{R}):x\mapsto B(x,-)$ and $Y\rightarrow L(X,\mathbb{R}):y\mapsto B(-,y)$ are injective.
Let $T:X\rightarrow X$ be a continuous linear transformation. Then, does there exist a continuous linear transformation $T^*:Y\rightarrow Y$ such that $B(Tx,y)=B(x,T^*y)$ for all $(x,y)\in X\times Y$?

REPLY [3 votes]: Just came across this old post. This is an excellent question and I came across it in my own research!
The answer is yes. The generalization is known as Singer's representation theorem, which was originally published in Russian in 1957.
I. Singer, "Linear functionals on the space of continuous mappings of a compact Hausdorff space into a Banach space", Rev. Math. Pures Appl. 2 (1957), pp. 301--315.
Just like the Riesz representation theorem, the continuity (more generally, the Hausdorff condition) and a boundedness assumption ensure the existence of the representation. An inf-sup condition ensures the uniqueness of the representation.
See e.g. the following papers in English for its proof and its further generalizations.

Wolfgang Hensgen, "A simple proof of Singer's representation theorem", Proc. Am. Math. Soc. 124, 10 (1996), pp. 3211--3212. (https://www.ams.org/journals/proc/1996-124-10/S0002-9939-96-03493-4/S0002-9939-96-03493-4.pdf)
A De Korvin and RJ Easton, "Some representation theorems", Rocky Mt. J. Math. 1, 3 (1971), pp. 561--573. (https://projecteuclid.org/download/pdf_1/euclid.rmjm/1250131827)