A formulation of continuity for bilinear forms

In summary, the conversation discusses the conditions for continuity in a bilinear form. The first two conditions, that the form is linear in both arguments and that there exists a constant M such that |a(x,y)|<M||x||||y||, imply continuity. However, the question arises as to whether continuity also implies the second condition. It is then mentioned that there is a theorem stating that a linear functional on a Hilbert space is continuous if and only if it is bounded. The conversation concludes that in this case, continuity does imply the second condition, as the norm of (x,y) must be less than 1 for the bilinear form to be bounded.
  • #1
quasar987
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[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.
 
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  • #2
I'm not quite sure what the question is. It seems to have been omitted. But just looking at your argument, z_n=(x_n,y_n)->0 doesn't imply x_n->0 or y_n->0. Does it?
 
  • #3
Well, I'm making use of the fact that if (M,d) is metric space, then the product topology on M x M is generated by the metric

D((x1,y1),(x2,y2))=[d(x1,x2)² + d(y1,y2)²]^½

I conclude that the norm on H x H is

||(x,y)|| = [||x||² + ||y||²]^½

And now if (x_n,y_n)-->0, this means that

[||x_n||² + ||y_n||²]^½ --> 0,

which can only happen if x_n-->0 and y_n-->0.

------------

You're right, I have not actually typed the question entirely. It is because I was confused by the fact that they seem to imply that condition (ii) is equivalent to continuity. While (ii) implies continuity, does continuity implies (ii)?
 
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  • #4
Well, there is the theorem that a linear functional on a Hilbert space is continuous if and only if it's bounded...
 
  • #5
You mean bounded in the unit ball?

In this case, you're right, it works. Because ||(x,y)|| = [||x||² + ||y||²]^½ <1 ==> ||x||, ||y||<1 and (ii) implies|a(x,y)|<M for all (x,y) in H x H such that ||(x,y)||<1.
 

Related to A formulation of continuity for bilinear forms

1. What is a bilinear form?

A bilinear form is a mathematical function that takes two inputs and produces a single output. It is linear in each of its arguments, meaning that the function satisfies the properties of additivity and homogeneity.

2. How is continuity defined for bilinear forms?

Continuity for bilinear forms is defined in terms of boundedness and convergence. A bilinear form is continuous if it is bounded and the inputs converge to a limit point, the output also converges to a limit.

3. What is the importance of continuity in mathematics?

Continuity is important in mathematics because it allows us to make predictions and draw conclusions about functions and their behavior. It is a fundamental concept in calculus and analysis, and is used to study the properties of functions and their derivatives.

4. How does the formulation of continuity for bilinear forms differ from other forms of continuity?

The formulation of continuity for bilinear forms is unique because it takes into account the behavior of the function in two separate directions, rather than just one. This allows for a more comprehensive understanding of the function and its properties.

5. Are there any practical applications of continuity for bilinear forms?

Yes, continuity for bilinear forms has many practical applications in fields such as physics, engineering, and economics. It is used to model and analyze various physical systems and processes, and is essential for understanding the behavior of these systems over time.

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