surjective and bijective mapping

[Bernoulli]

Hi, can anyone tell me what a surjective mapping between Hilbertspaces is? That is: what does surjective mean? What about bijective?

I mean what is special about a mapping if it is sujective or bijective?

[matt grime]

A map f:X -> Y is

injective if f(x)=f(z) => x=z, ie for any point in the image there is a unique preimage.

surjective if for all y in Y there is an x in X such that f(x)=y

bijective if it is both injective and surjective


A map has an inverse iff it is bijective.

I don't understand how you've got to Hilbert Spaces without being taught this.

[Bernoulli]

Thanks for fast reply...

I must have been away when they told us that.

[matt grime]

It's just that the definition of inj and surj and hence bijection is 1st year undergrad maths, if not school, and hilbert spaces is 2nd or 3rd year undergraduate maths.

[HallsofIvy]

The terms "one-to-one" and "onto" are sometimes used for "injective" and "surjective".

A function from one set to another (doesn't have to be a Hilbert Space) is "injective" or "one-to-one" if and only if f(x)= f(y) implies x= y. In other words, only one value of x gives any one value of y.

A function from one set to another is "surjective" or "onto" if and only if for every y in the range set, there exist x in the domain such that f(x)= y. In other words, there are no "left over" members of the range set.

[Bernoulli]

Im in my third year now, and i never really heard the formal definition on this before. I came across the words in a book and i just wondered what they ment.

But anyway, this seams like a very good site.