Continuity of bilinear maps Given a vector space $V$ over $\mathbb{R}$ with a norm $||*|| $. Can $(x,y)\rightarrow(x+y)$ be an example of continous bilinear map, if yes, can you please exlain why?
Definition of continuous bilinear map $\lambda$ on $V\times V \rightarrow V$ is:
For all $v,w\in V$, there is $C>0$ such that: $||\lambda(v,w)||\le C||v||||w||$, how can I proceed from here?

Your map $\lambda:V\times V\rightarrow V$ is continuous, but not bilinear:
For $\mu\not=0\in \mathbb{R}$ and $v,w\not=0\in V$:
$$\lambda(\mu v,w)=\mu v+ w\not = \mu (v+w)=\mu\cdot\lambda(v,w)$$
However, $\lambda$ is a linear map from the vector space $V\times V$ to $V$. Therefore it is continuous if and only if there exists $C>0$ such that
$$\|\lambda(x,y)\|_V \le C \|(x,y)\|_{V\times V}\tag{1}$$
Note that $\|x\|_V \|y\|_V$ is not a norm on $V\times V$. A suitable norm (where I mean by suitable that it generates the product topology induced by $\|\cdot\|_V$) would for example be
$$\|(x,y)\|_V = \|x\|_V + \|y\|_V$$
With respect to that norm, $\lambda$ clearly satisfies $(1)$ with $C=1$ by the triangle inequality.
