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By Dawson C.N., Martinez-Canales M.L.

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A? 4. 16. Let H0 ˚ H1 be a direct sum of two (complex) Hilbert spaces and let A Â H0 ˚ H1 be a relation. Then LinC A D A : Proof. 6) and by definition of the adjoint relation we have A D . A D 1 ? / / . A?? / 1 1 ? / / 1 / D A?? D LinC A: Here we have used . 1/. 4 holds. 17. Note that A is indeed a double ortho-complement. 4 applies. So, if A is a linear relation then its (strong) closure AN is also equal to its weak closure. If AN is a linear mapping then it can be also characterized as the N adjoint relation of A .

C / ! C /; x 7! CnC1;n x for n 2 Z. 8. C //n2Z be the Sobolev chain associated with the operator C  H ˚ H . C / ! C /; x 7! C / for every k 2 Z. 0/ are unitarily equivalent; k 2 Z: Proof. C /. 0/ D C , then equality of resolvent sets, indeed of all spectral parts, follows. C / ! C / for all j; k 2 Z. C /: For k 2 Z<0 we may argue in the following way. C /. C /N such that xj ! C / and kC1 xj ! C /: for j ! 1. 0/ xj ! 0/ xj ! C /; for j ! 0/ Ck;0 y D z1 . In particular, we have that xj ! C / and so xj !

2 Some Construction Principles of Hilbert Spaces 25 for all t0 2 M0 . tn / D 0 for all ti 2 Mi ; i D 0; : : : ; n: i This tensor product construction is extremely useful for our context, since it allows for the transition from one variable to multi-variable function spaces. 3. Z/ are complex number sequences, we realize that with Mi D Z we encounter an instance of the last proposition. i 7! zin /in 2Z extends to a linear isometry I . Z/. 4. R/ This result follows by the same arguments as the previous example.

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Acharacteristic-Galerkin Aproximation to a system of Shallow Water Equations by Dawson C.N., Martinez-Canales M.L.

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