$\langle \chi,\chi \rangle \ge 1$ for any representation Let G be a finite group and $\chi$ its character.
If degree (or dimension? I am not sure about terminology) of $\chi$ is $> 1$, is it always true that $\langle\chi,\chi\rangle \ge 1$?
My intuition is that this quantity is the sum of squares of dimensions of sub-representations of $\chi$.
So as long as we don't know if $\chi$ is irreducible, it should at least contain the trivial representation so $\langle\chi, \chi\rangle \ge 1$.
Does this make sense?
Thank you for your help. 
 A: Let $G$ be a finite group, with $\rho : G \to \operatorname{GL}(V)$ some  representation of $G$ for $V$ some finite dimensional vector space over some field $\mathsf{k}$ of some characteristic not dividing $\left| G \right|$ (feel free to assume $\mathsf{k} = \mathbb{C}, \mathbb{R}, \mathbb{Q}$). Then Maschke's Theorem states that $\rho$ is completely reducible. What does this mean? Well, if $U \leq V$ is a subrepresentation of $V$, then there exists some other subrepresentation $W \leq V$ satisfying $V = U \oplus W$. Since $V$ is finite dimensional, we can repeat this step with $U,W$ until we are left with a decomposition of the form $V = \bigoplus_{i=1}^{k} V_i$ for some subrepresentations $V_i \leq V$ that have no proper subrepresentations, i.e are irreducible. Then
$$
\operatorname{Hom}_{\mathsf{k}\left[G\right]}(V,V) \cong \bigoplus_{i,j} \operatorname{Hom}_{\mathsf{k}\left[G\right]}(V_i,V_j).
$$
Then let $\chi$ denote the character of $\rho$, and $\chi_k$ denote the character of the restriction of $\rho$ to $V_k$. Then 
$$
\langle \chi, \chi \rangle = \dim_{\mathsf{k}} \operatorname{Hom}_{\mathsf{k}\left[G\right]}(V,V) = \sum_{i,j} \dim_{\mathsf{k}} \operatorname{Hom}_{\mathsf{k}\left[G\right]}(V_i,V_j) = \sum_{i,j} \langle \chi_i, \chi_j \rangle.
$$
Now we will write $V \cong \bigoplus_{i=1}^{\ell} n_i V_i$, for some integers $n_i$, and $V_i \not \cong V_j$ when $i \neq j$, and assume now that $\mathsf{k}$ is algebraically closed. Then by Schur's Lemma 
$$
\dim_{\mathsf{k}} \operatorname{Hom}_{\mathsf{k}\left[G\right]}(V_i,V_j) = \delta_{i,j},
$$ 
where $\delta_{i,j}$ denotes the kronecker delta function. Thus
$$
\langle \chi, \chi \rangle  = \sum_{i,j} n_i n_j \delta_{i,j} = \sum_{i,j} n_i^{2}.
$$ 
Thus $\langle \chi, \chi \rangle$ is a sum of squared integers, and so $\langle \chi, \chi \rangle \geq 1$, with equality if and only if $\rho$ was irreducible to being with.

A few comments. When you say "Let $G$ be a finite group and $\chi$ its character", what you mean is "a character". A finite group will have many different characters, infinitely many in fact. What you wish to say is "Let $G$ be a finite group, with $\chi$ the character of some representation $\rho$ of $G$. Also, as others have pointed out, reducible representations need not contain a copy of the trivial representation. A contrived example would be the character of the representation 
$$
\rho : C_2 = \langle s \mid s^{2} \rangle \to \operatorname{GL}(\mathbb{C}^{2}) \ : \ s \mapsto \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}. 
$$
Then $\rho$ is the direct sum of two copies of the sign representations of $C_2$, so is reducible and does not contain any copies of the trivial representation. 
