Assume $\phi:\mathbb{H}_1\rightarrow \mathbb{H}_2$ is a continuous linear mapping between two real Hilbert spaces $\mathbb{H}_1$ and $\mathbb{H}_2$. If $\mathbb{H}_2=\mathbb{R}$, then the Riesz representation theorem entails that there exists $\xi \in \mathbb{H}_1$ such that $\phi(x)=\langle x,\xi\rangle_{\mathbb{H}_1}$, $\forall x\in\mathbb{H}_1$.

Can the usual Riesz representation be extended to the general case where $\mathbb{H}_2$ is not necessarily $\mathbb{R}$? If not, is there any practical way to characterize the functional $\phi$? I am looking for references addressing any of these points.

closed as off-topic by Nik Weaver, Stefan Waldmann, Yemon Choi, paul garrett, David Handelman Nov 29 at 1:36

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  • Consider the cases $H_1 = H_2 = \mathbb{R}^2$ or $H_1 = H_2 = \mathbb{R}^3$. What kind of answer would you want there? – Nik Weaver Nov 27 at 11:47
  • @epsilone We can define a map $\langle \cdot, \cdot \rangle : \mathbb{H}_1 \times L(\mathbb{H}_1,\mathbb{H}_2) \rightarrow \mathbb{H_2}$ by $\langle x, f \rangle = f(x)$. Then a "Riesz representation theorem" holds, almost tautologically. Is this this what you want? I think you need to give a more of a reason for considering this question. – Robert Furber Nov 27 at 11:54
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    @epsilone A topological tensor product defined using a norm can never fill up all of $L(\mathbb{H}_1,\mathbb{H}_2)$ when $\mathbb{H}_1$ and $\mathbb{H}_2$ are both infinite-dimensional. The largest Banach space (therefore with the smallest norm) tensor product is the injective tensor, which for Hilbert spaces corresponds to compact operators as described in user131781's answer. – Robert Furber Nov 27 at 15:46
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    The continuous linear operators from $H_1$ to $H_2$ which correspond to elements of the Hilbert space tensor product are called "Hilbert-Schmidt operators". This is a subset of the compact operators. – Nik Weaver Nov 27 at 15:59
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    "The intuition is clear, what I do not see so clear are the specifics" - well unfortunately, this means what you are looking for is not a reference request, but a source from which to learn the details of functional analysis. I suggest that in this case, MSE would be a more suitable site – Yemon Choi Nov 27 at 21:59

Your question is rather vague but one result that might be of interest is the singular value decomposition theorem which states that every compact operator can be written in the form $$T(x)=\sum \lambda_n (x|x_n)y_n$$ where $(\lambda_n)$ is a sequence of scalars which converges to $0$ and the other two sequences are orthonormal bases for the appropriate Hilbert spaces (which I am assuming to be separable for simplicity). This is a simple consequence of the spectral theorem for compact, self-adjoint operators. Both of these results can be found in virtually any text on Hilbert space or elementary introduction to functional analysis. Googling „Singular value decomposition for compact operators“ will turn up a number of self-contained approaches.

Adding more information based on comments. The connection with the Riesz theorem is, of course, that the latter is the case where the image space is one-dimensional. As pointed out above, one obtains many special classes of operator by assuming that the sequence of scalars is from a suitable sequence space, most notably an $\ell^p$-space ($p=1$ and $p=2$ give the nuclear resp. Hilbert Schmidt classes). Again, as mentioned above, these can be interpreted using tensor products. The class of all continuous linear operators cannot be obtained as a tensor product of Banach spaces (again noted in the comments) but it is a tensor product in the sense of locally convex spaces (Grothendieck) when we regard the Hilbert spaces as complete lcs‘s using the topology of uniform convergence on compact subsets.

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