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