It is an easy question, but i want to make it clear :)
Let $(V,\langle -,- \rangle)$ be an inner product space over $\mathbb{K}$.
Then, is the inner product $\langle -,- \rangle:V\times V\rightarrow \mathbb{K}$ continuous?
Indeed, I have proven it, but I want to make it sure :)
Is this true?
EDIT:
Here's what i tried.
I tried to show that $\langle -,- \rangle$ is continuous at a fixed point $(\alpha,\beta)$.
Assume that for a given $\epsilon >0$, there is a neighborhood $N$ of $(\alpha,\beta)$ such that for all $(x,y)\in N, |\langle x,y \rangle - \langle \alpha,\beta \rangle |<\epsilon$.
Let $\| \cdot \|$ be the norm induced by the inner product.
Then, it is only sufficient to show that $\| x-\alpha \| \| y \| + \| \alpha \| \| y - \beta \|$ is smaller than $\epsilon$.
Consider separate cases when either $\alpha$ or $\beta$ is zero.