Question on meaning of some symbols

[yungman]

I don't know the meaning of these:


1) $$sup_{B_\delta}|f(x,y)|$$

Where $B_\delta$ is the ball of radius $\delta$.

2) $$\int \int _{R^2 \B _{\delta} } f(xy)dxdy$$

I don't know what is $R^2$\B$_{\delta}$

Please read my latex because the symbol really don't show correctly.

 

[Landau]

1) The supremum of {|f(x,y)|} where (x,y) ranges over the ball centered at 0 of radius delta: |(x,y)|=sqrt(x^2+y^2)<delta.

2) the plane R^2 without the ball centered at 0 of radius delta, i.e. \ (in Latex: "\backslash") means 'complement' or 'set difference'. So it consists of pairs (x,y) of real numbers such that |(x,y)|=sqrt(x^2+y^2)>=delta.

 

[yungman]

1) The supremum of {|f(x,y)|} where (x,y) ranges over the ball centered at 0 of radius delta: |(x,y)|=sqrt(x^2+y^2)<delta.

2) the plane R^2 without the ball centered at 0 of radius delta, i.e. \ (in Latex: "\backslash") means 'complement' or 'set difference'. So it consists of pairs (x,y) of real numbers such that |(x,y)|=sqrt(x^2+y^2)>=delta.

Thanks for you reply, so for

1) Is the upper bound of |f(x,y)| in the ball.

2) Is the whole 2D plane minus the circle center at 0 with radius $\delta$

 

[Landau]

1) Is the upper bound of |f(x,y)| in the ball.

The least upper bound, a.k.a. the supremum ;)

2) Is the whole 2D plane minus the circle center at 0 with radius $\delta$

Yes.

 

[HallsofIvy]

Thanks for you reply, so for

1) Is the upper bound of |f(x,y)| in the ball.

No. There is no such thing as "the" upper bound of a set of numbers. If a set has an upper bound, then it has an infinite number of upper bounds. This is the least upper bound- the smallest number in the set of all upper bounds.

2) Is the whole 2D plane minus the circle center at 0 with radius $\delta$