The Orthogonal Decomposition Theorem
Let $V$ be a Hilbert space and $M\subset V$ a closed subspace of $V$. Then (i) $M^\bot$ is a closed subspace of $V$. (ii) $V$ can be represented as the direct sum of $M$ and its orthogonal complement $M^\bot$
$$ V=M\oplus M^\bot $$i.e., every vector $v\in V$ can be uniquely decomposed into two orthogonal vectors $\boldsymbol m$, $\boldsymbol n$, s.t.
$$ \boldsymbol v=\boldsymbol m+\boldsymbol n, \boldsymbol m\in M,\boldsymbol n\in M^\bot $$COROLLARY Let $V$ be a Hilbert space and $M$ a vector subspace of $V$ . The following conditions are equivalent to each other (i) $M$ is closed. (ii) $(M^\bot)^\bot = M$.
Orthogonal projection
Let $M$ be a closed subspace of a Hilbert space $V$. The linear projection $P_M$ corresponding to the decomposition
$$ V=M\oplus M^\bot, \boldsymbol v=\boldsymbol m+\boldsymbol n $$$$ P_M:V\to V, P_M\boldsymbol v=\boldsymbol m $$is called the orthogonal projection onto the subspace $M$ s.t. $\parallel P_M\parallel=1$ (unit norm).
Riesz Representation Theorem
$$ f(\boldsymbol v) = (\boldsymbol v,\boldsymbol u),\;\forall \boldsymbol v\in V $$$$ \| f\|_{V\prime} =\|\boldsymbol u\|_V $$
COROLLARY
$$ \langle R\boldsymbol u,\boldsymbol v\rangle= (\boldsymbol u,\boldsymbol v),\;\forall \boldsymbol u,\boldsymbol v\in V $$$$ \|R\boldsymbol u\|_{V\prime} = \|\boldsymbol u\|_V $$In particular, in the case of a real Hilbert space $V$ , $R$ is a linear norm-preserving isomorphism (surjective isometry) from $V$ onto $V\prime$. $R$ is also known as Riesz map to identify the topological dual of $V$ with itself.
$$ R_{V\prime}:V\prime\ni g\to\{f\to(f,g)_{V\prime}\in\mathbb{C}\}\in(V\prime)\prime $$where$(V\prime)\prime$ is the bidual of $V$.
Adjoint of a Continuous Operator
$$ (\alpha_1A_1 + \alpha_2A_2)^\ast =\overline\alpha_1A^\ast_1 + \overline\alpha_2A^\ast_2 $$$$ (B\circ A)^\ast = A^\ast\circ B^\ast $$$$ (id_U )^\ast = id_U $$$$ (A^\ast)^{−1} = (A^{−1})^\ast $$$$ \|A\|_{\mathcal L(U,V )} = \|A^\ast\|_{\mathcal L(V,U)} $$$$ (A^\ast)^\ast = A $$
Symmetric and Self-Adjoint Operators
$$ D(A)\subset D(A^\ast) \;and\; A^\ast|_{D(A)} = A $$
If, additionally, the domains of both operators are the same, i.e., $A = A^\ast$, then we say that operator $A$ is self-adjoint. Obviously, every self-adjoint operator is symmetric, but not conversely. In the case of a continuous and symmetric operator $A$ defined on the whole space $U$, however, the adjoint $A^\ast$ is defined on the whole $U$ as well, and, therefore, $A$ is automatically self-adjoint.
Example
$$ \boldsymbol u=\boldsymbol u_1+\boldsymbol u_2\; where\; \boldsymbol u_1\in M,\boldsymbol u_2\in M^\bot $$$$ (P\boldsymbol u,\boldsymbol v)=(\boldsymbol u_1,\boldsymbol v)=(\boldsymbol u_1,\boldsymbol v_1 +\boldsymbol v_2)= (\boldsymbol u_1,\boldsymbol v_1) = (\boldsymbol u_1 +\boldsymbol u_2,\boldsymbol v_1) = (\boldsymbol u,\boldsymbol v_1) = (\boldsymbol u,P\boldsymbol v) $$for every $\boldsymbol u, \boldsymbol v\in V$.
Some norms in Soblev Spaces
$$ H^1(\Omega) :=\{u\in L^2(\Omega) : \dfrac{\partial u}{\partial x_i} \in L^2(\Omega), i = 1,\;\cdots,N\} $$$$ \|u\|_{H^1}^2 :=\|u\|^2 + \sum_{i=1}^N\|\dfrac{\partial u}{\partial x_i}\|^2 $$$$ |u|^2_{H_1}:= \sum_{i=1}^N \|\dfrac{\partial u}{\partial x_i}\|^2 $$$$ H(div, \Omega) := \{\boldsymbol\sigma = (\sigma_i)^N_{i=1} \in (L^2(\Omega))^N : div\;\boldsymbol\sigma \in L^2(\Omega)\} $$$$ \|\boldsymbol\sigma\|^2_{H(div)} := \|\boldsymbol\sigma\|^2 + \|div\;\boldsymbol\sigma\|^2 $$$$ \|\boldsymbol\sigma\|^2 := \sum_{i=1}^N \|\sigma_i\|^2 $$
Example
$$ (\boldsymbol\sigma,\nabla u)=−(div\;\boldsymbol\sigma,u)+\langle\sigma_n,u\rangle, \boldsymbol\sigma\in H(div,\Omega),u\in H_1(\Omega) $$The brackets $\langle\cdot,\cdot\rangle$ denote the duality pairing between $H^{1/2}(\Gamma)$ and $H^{−1/2}(\Gamma)$ (topological dual of each other) that generalizes the usual integral over the boundary.
$$ H^1_{\Gamma_1} (\Omega):=\{u\in H^1(\Omega) : u=0\;on\;\Gamma_1\} $$$$ H_{\Gamma_2}(div,\Omega):=\{\boldsymbol\sigma\in H(div,\Omega) : \sigma_n =0\;on\;\Gamma_2\} $$$$ (\boldsymbol\sigma,\nabla u)=−(\nabla\cdot\boldsymbol\sigma,u)\;\boldsymbol\sigma\in H_{\Gamma_2}(div,\Omega),u\in H_{\Gamma_1}^1 (\Omega) $$Variational formulation for diffusion-convection-reaction eqaution
$$ \left\{\begin{array}{ll} u\in H^1_{\Gamma_1}(\Omega) \\[2ex] \displaystyle{\int_\Omega}(a_{ij}u_{,j}v_{,i} − b_iuv_{,i} + cuv) = {\int_\Omega} fv \;\;\;v\in H_{\Gamma_1}^1 (\Omega) \end{array}\right. $$
Note: The convection term is also integrated by parts here to avoid integrable requirement for high-order derivative of $b$.
Poincaré Inequality
$$ c\|u\|^2 \leq |u|^2_{H^1} $$for all $u \in H^1(\Omega)$ that vanish on $\Gamma_1$.