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From pp.20 in the book "A Fenics Tutorial" by Logg we see the following passage:

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My question is: What does the [V]d mean? What do square brackets around a vector space mean? Is there a reference for this?

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  • $\begingroup$ Sounds to me as if: If $V$ is the space of (continuous, differentiable, smooth, ...) functions $\Omega\to \Bbb R$, then $[V]^d$ shall denote the space of (continuous, differentiable, smooth, ...) functions $\Omega\to \Bbb R^d$. $\endgroup$ – Hagen von Eitzen Aug 9 '17 at 21:42
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Let $V$ be a function space of functions $u: \Omega \to \mathbb{R}$. For example $V=C(\Omega)$, $L^2(\Omega)$, $H^1(\Omega)$. Note that one can also write $L^2(\Omega;\mathbb{R})$ to explicitly indicate that the functions map to $\mathbb{R}$.

Then there are lot of different notations in literature to indicate the spaces $C(\Omega;\mathbb{R}^d)$, $L^2(\Omega;\mathbb{R}^d)$, $H^1(\Omega;\mathbb{R}^d)$, which are spaces of vector-valued functions $u:\Omega \to \mathbb{R}^d$.

One sees for example $L^2(\Omega;\mathbb{R}^d), L^2(\Omega)^d, \{L^2(\Omega)\}^d, \mathbf{L}(\Omega), \mathbb{L}(\Omega), [L(\Omega)]^d$. You asked about the last one. Here are some references where they use exactly that notation:

  • Arndt, Daniel, Helene Dallmann, and Gert Lube. "Local projection FEM stabilization for the time‐dependent incompressible Navier–Stokes problem." Numerical Methods for Partial Differential Equations 31.4 (2015): 1224-1250.
  • Du, Qiang, and Max D. Gunzburger. "Analysis of a Ladyzhenskaya model for incompressible viscous flow." Journal of Mathematical Analysis and Applications 155.1 (1991): 21-45.
  • Feireisl, Eduard, Antonın Novotný, and Hana Petzeltová. "On the existence of globally defined weak solutions to the Navier—Stokes equations." Journal of Mathematical Fluid Mechanics 3.4 (2001): 358-392.
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