Is every bilinear mapping bounded?

[yifli]

In a book I'm reading, it defines a bounded bilinear mapping \omega: X\times Y\rightarrow W, where X,Y and W are all normed linear spaces as
\left\| \omega(\xi,\eta)\right\| \leq b \left\| \xi \right\| \left\| \eta \right\|

So it uses \left\| \xi \right\| \left\| \eta \right\| as a norm on the product space.

Is this a valid norm? I can't prove it is equivalent to the norm \left\| \xi \right\| + \left\| \eta \right\|

 

[yifli]

In a book I'm reading, it defines a bounded bilinear mapping \omega: X\times Y\rightarrow W, where X,Y and W are all normed linear spaces as
\left\| \omega(\xi,\eta)\right\| \leq b \left\| \xi \right\| \left\| \eta \right\|

So it uses \left\| \xi \right\| \left\| \eta \right\| as a norm on the product space.

Is this a valid norm? I can't prove it is equivalent to the norm \left\| \xi \right\| + \left\| \eta \right\|

I guess I was in the wrong direction:
a bounded linear mapping T is defined as \left\|T(\xi) \right\| \leq b \left\|\xi\right\|
but this definition cannot be applied to bilinear mapping. Am I correct?

That being said, the boundedness of a bilinear mapping may be shown as follows:

The bilinear mapping \omega: X\times Y\rightarrow W is equivalent to the linear mapping T: X \rightarrow Hom(Y,W). Since X, Y and W are finite-dimensional normed linear space, T is bounded and any mapping in Hom(Y,W) is bounded. Now I can show that the bound b of the bilinear mapping is actually the bound of T:
\left\| \omega(\xi,\eta) \right\| =\left\|T(\xi)(\eta) \right\| \leq \left\| T \right\| \left\| \xi \right\| \left\| \eta \right\|