I have a question about these definitions:
Let $d \geq 0$ be a dimension. We let $\{0, 1\}^d$ be the standard
discrete d-dimensional cube, consisting of d-tuples $\omega=(\omega_1,\dots,\omega_d)$ where $\omega_j \in \{0, 1\}$ for $j = 1, \dots,d$. If $h=(h_1,\dots,h_d) \in \mathbb{Z_N^d}$ we define $\omega \cdot h := \omega_1h_1+\dots+\omega_dh_d$. If $(f_\omega)_{\omega \in \{0,1\}^{d}}$ is a $\{0,1\}^{d}$ d-tuple of functions in $L^{\infty}(\mathbb{Z_N})$ we define the d-dimensional Gowers inner product by the formula: $[(f_\omega)_{\omega \in \{0,1\}^{d}}]_{U^{d}}:=\mathbb{E}\bigg(\prod_{\omega \in \{0,1\}^{d}} C^{|\omega|}f_\omega(x+\omega \cdot h) \bigg | x \in \mathbb{Z_N}, h \in \mathbb{Z_N^d}\bigg)$. And for a function $f:\mathbb{Z_N} \to \mathbb{C}$ we define the Gowers uniformity norm by: $\lVert f \rVert_{U^{d}}=[(f)_{\omega \in \{0,1\}^{d}}]_{U^{d}}^{1/2^d}$.
Now for a function $f:\mathbb{Z_N} \to \mathbb{C}$, i want to compute
$[(f)_{\omega \in \{0,1\}^{1}}]_{U^{1}}=\mathbb{E}(f(x)\overline{f(x+h)}|x,h \in \mathbb{Z_N})=\frac{1}{N^2} \sum_{x \in \mathbb{Z_N},h \in \mathbb{Z_N}} f(x)\overline{f(x+h)}$. Why this quantity is equal to $|\mathbb{E}(f(x)|x \in \mathbb{Z_N})|^{2}$? (!)
If it was true, $\lVert f \rVert_{U^{1}}=|\mathbb{E}(f)|$. So for $d=1$ Gowers uniformity norm is not a real norm, because i can find an example of a function $f$ in $\mathbb{Z_N}$ not null, but $|\mathbb{E}(f)|=0$ (for example with $N$ even, $f=1$ on odd numbers and $-1$ on even numbers, so $\frac{1}{N}\sum_{x \in \mathbb{Z_N}}f(x)=\mathbb{E}(f)=0)$.
My problem is in (!) and can the final example go well?
Yes, you are correct. When $d=1$, by definition,
$$ \| f\|_{U^1} = \lvert \mathbb{E}_x f(x)\rvert.$$
As you point out, this is not a norm, but only a seminorm (in that it is non-negative, homogenous, and obeys the triangle inequality). When $d\geq 2$ the Gowers uniformity norm is, indeed, a norm in the usual sense.