- #1

- 4,807

- 32

**[SOLVED] A formulation of continuity for bilinear forms**

## Homework Statement

My HW assignment read "Let H be a real Hilbert space and a: H x H-->R be a coninuous coersive bilinear form (i.e.

(i) a is linear in both arguments

(ii) There exists M>0 such that |a(x,y)|<M||x|| ||y||

(iii) there exists B such tthat a(x,x)>a||x||^2"

So apparently, condition (ii) is the statement about continuity. But I fail to see how this statement is equivalent to "a is continuous".

I see how (ii) here implies continuous, but not the opposite.

## The Attempt at a Solution

Let z_n = (x_n,y_n)-->0. Then x_n-->0 and y_n-->0. So |a(x,y)|<M||x|| ||y|| implies a(x,y)-->0. a is thus continuous at 0, so it is so everywhere, being linear.

Last edited: