I have proven something that is definitely not true (Lemma 2), which is why I am intersted where I err.
Definition Let $C(\mathbb{R}/\mathbb{Z},\mathbb{C})$ be the set of all continuous $\mathbb{Z}$-periodic functions from $\mathbb{R}$ to $\mathbb{C}$. We define inner product by $$\langle f,g\rangle = \int_{[0,1]}f\bar{g}$$
and $L^2$ norm by $$\|f\|_2 = \sqrt{\langle f,f \rangle}$$
Lemma 1. I have already proven all the properties of inner product, such as (a) positivity (b) hermitian property (c) linearity in the first variable and antilinearity in the second.
Lemma 2. I have to demonstrate the Cauchy-Schwarz inequality
$$|\langle f,g \rangle | \leq \|f\|_2 \cdot \|g\|_2$$
But somehow magically I managed to show that they are always equal
$\begin{array}{ll}
\iff \sqrt{\langle f,g \rangle \bar{\langle f,g \rangle }} = \sqrt{\langle f,f \rangle \langle g,g \rangle } \quad \text{by definition}\\
\iff \sqrt{\langle f,g \rangle \langle g,f \rangle } = \sqrt{\langle f,f \rangle \langle g,g \rangle } \quad \text{hermitian property}\\
\iff \sqrt{ \langle \langle f,g \rangle g,f \rangle } = \sqrt{\langle f\langle g,g \rangle,f \rangle } \quad \text{linearity}\\
\iff \sqrt{ \langle \langle fg,g \rangle,f \rangle } = \sqrt{\langle \langle fg,g \rangle,f \rangle } \quad \text{linearity}\\
\end{array}$
Where does my mistake dwell? Also, any help on how to proceed would be appreciated.
Best Answer
Your mistake is the last step. Linearity allows a scalar $k \in \Bbb{F}$ enters into the bracket $\langle u,v \rangle$ $$k\langle u,v \rangle = \langle ku,v \rangle,$$ but not a vector $w \in V$ $$w\langle u,v \rangle = \langle wu,v \rangle. \tag{wrong!}$$
Here, the $V = C(\mathbb{R}/\mathbb{Z},\mathbb{C})$, $\Bbb{F} = \Bbb{C}$, $u = f$ and $v = g$.