Why of this bound $||u_{m0}|| \leq ||u_0||$, where $u_{m0}$ is a projection of $u_0$?

Question

I'm dealing with Navier-Stokes equations, using the book of Teman. Theres a bound of a projection of a function that i didn't understood, so i will introduce the main concepts with i am dealing with.

Consider $u_m=\sum_{i=1}^m g_{im}(t)w_i$ an approximation of $u$ via galerkin method, where $w_i$ are the basis and $g_{im}(t)$ the coeficients, $u_m(0)=u_{m0}$ and $u(0)=u_0$.

The space $V$ is basecally the closure, in $H^1_0(\Omega)$, of the set of all function $C^\infty$ with support on a open set $\Omega \subset \mathbb{R}^2$, with it's divergent being null.

Let $\{w_j\}$ be a base of $V$ which we can take $u_{m0}$ as a projection in $V$ of $u_0$ spanned by $w_1,w_2, \cdots,w_m $.

So, why this bound is valid, $||u_{m0}|| \leq ||u_0||$?

$||\cdot|| $ is the norm in $H^1_0$.

Answer 1

Usually, in the context of Galerkin method, "base" of a separable normed space $X$ means a linearly indpendent countable set whose span is dense in $X$ (the existence of this family follows from the separability - see Lemma 4.1, p. 83, in Le Dret). On the other hand, in the context of Hilber spaces, "base" usually means a maximal orthonormal set which is characterized as follows:

Theorem. Let $\{e_j\}_{j\in J}$ be an orthonormal family in a Hilbert space $H$. The following assertions are equivalent:

• $\{e_j\}_{j\in J}$ is maximal.
• $ x=\sum_{j\in J}(x,e_j)e_j$ for all $x\in \mathcal{H}$.
• $ \|x\|^2=\sum_{j\in J}|(x,e_j)|^2$ for all $x\in \mathcal{H}$.
• $\overline{\operatorname{span}\{e_j\mid j\in J\}}=\mathcal{H}$.

Proof: See Corollary 12.8, p. 532, in Knapp and Proposition 2.3, p. 479, in Taylor.

From this theorem, if $X$ is Hilbert separable, then a base in the "Hilbert space sense" is also a base in the "Galerking sense".

Therefore, assuming that $(w_j)_{j\in\mathbb N}$ is base of $V$ (which is Hilbert separable) in the "Hilbert space sense", using the above theorem we conclude that

$$\begin{aligned} \|u_{m0}\|^2&=(u_{m0},u_{m0})\\ &=\left(\sum_{i=1}^m(u_0,w_i)w_i,\sum_{j=1}^m(u_0,w_j)w_j\right)\\ &=\sum_{i=1}^m(u_0,w_i)\sum_{j=1}^m\overline{(u_0,w_j)}\left(w_i,w_j\right)\\ &=\sum_{i=1}^m|(u_0,w_i)|^2\left(w_i,w_i\right)\\ &=\sum_{i=1}^m|(u_0,w_i)|^2\\ &\leq \sum_{i=1}^\infty|(u_0,w_i)|^2\\ &=\|u_0\|^2 \end{aligned}$$