I am reading the book of Mawhin and Willem on critical point theory and Hamiltonian systems and I am currently reading about functionals with symmetries. However, I am a bit puzzled about the invariance of some subset under a group representation. I will first provide the relevant definitions.
Let $G$ be a topological group. A representation of $G$ over a Banach space $X$ is a family $\{T(g)\}_{g\in G}$ of linear operators $T(g):X\to X$ such that $$T(0)=\operatorname{Id},$$ $$T(g_1+g_2)=T(g_1)T(g_2),$$ $$(g, u)\mapsto T(g)u \text{ is continuous. }$$ A subset $A$ of $X$ is invariant (under the representation) if $T(g)A=A$ for all $g\in G$. A representation $\{T(g)\}_{g\in G}$ of $G$ over $X$ is isometric if $||T(g)u||=||u||$ for all $g\in G$ and all $u\in X$.
Now let us consider for some $T>0$ the Sobolev space $$H^1_T=\{u\in [H^1(0, T)]^3 : u(0)=u(T)\}$$ of $T$-periodic functions in $H^1$. We take the representation of $S^1$ given by translation in time, i.e. $$(T(\theta)u)(t)=u(t+\theta).$$ We have the following result about isometric representations of $S^1$:
Let $\{T(\theta)\}_{\theta\in S^1}$ be an isometric representation of $S^1$ over $\mathbb{R}^N$ (with the Euclidean norm). Assume that $\operatorname{Fix}(S^1)=\{0\}$, where $$\operatorname{Fix}(S^1)=\{u\in \mathbb{R}^N : T(\theta)u=u, \forall \theta\in S^1\}.$$ Then $N$ must be even.
In our case, the representation is obviously isometric and the fixed points of the representation are the constant functions. Consider now the following subspace of $H^1_T$: $$V:=\left\{a\cos\left(\frac{2\pi t}{T}\right)|a\in \mathbb{R}^3\right\}.$$ We have $\operatorname{dim}(V)=3$ and $V\cap \operatorname{Fix}(S^1)=\{0\}$. Now I believe that $V$ is invariant under the representation $\{T(\theta)|_V\}_{\theta\in S^1}$, but if this were the case then $\operatorname{dim}(V)$ would be even, which is not true. So, what am I missing, am I not reading the result I mentioned properly or is $V$ not invariant under translation in time?