# Prove that this $\langle u,v\rangle = \int_\Omega [ \bigtriangledown u \cdot\bigtriangledown v +uv ]$ is an inner product in $H^1_0$

Information needed

$$I = (a,b)$$

$$H_0^1 (I) = \{ v \in H^1(I) : v(a)= v(b) = 0 \}$$

$$H^1 (I) = \{ v: v , v' \in \mathbb{L}_2(I) \}$$

$$\mathbb{L}_2(I) = \{ v:v \text{ it's defined at } I \text{ and } \int v^2\,dx < \infty \}$$

Prove that this is a scalar product in $$H^1_0$$

Using the notation $$\langle u,v\rangle = \int_\Omega [ \bigtriangledown u \cdot\bigtriangledown v +uv ]$$

where $$\Omega$$ is a bounded domain.

In the book numerical solutions of PDE by the finite element method by Claes there is this passage and I could not see clearly this statement.

Thanks to any help !

• Have you tried proving it yourself? An inner product has three defining properties, each of which should not be too hard to verify. – MaoWao Jan 30 at 12:15
• you mean, symmetry; linearity and podsitive definiteness ? – Saiten Jan 30 at 12:19
• It's bilinearity (linearity in each argument separately), not linearity. Otherwise yes. – MaoWao Jan 30 at 12:23
• Use $\langle X\rangle$ for $\langle X\rangle$. – Shaun Jan 30 at 12:39
• Thankss for the help! – Saiten Jan 30 at 13:24

1. $$\langle u,v \rangle = \langle v,u \rangle$$.
2. $$\langle a u, v\rangle = a \langle u,b\rangle$$ and $$\langle u+v,w\rangle = \langle u,w\rangle+\langle v,w\rangle$$.
3. $$\langle u,u\rangle\ge 0$$ and $$\langle u,u\rangle=0 \Leftrightarrow u = 0$$
The third property is the trickiest. You need to show that if $$u \in H^1_0$$ and $$\int_I (|\nabla u|^2 + u^2) = 0$$ then $$u=0$$ almost everywhere in $$I$$.
• Use $\langle X\rangle$ for $\langle X\rangle$. – Shaun Jan 30 at 13:17