# Surjective and bijective mapping

1. Dec 10, 2004

### 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?

2. Dec 10, 2004

### 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.

3. Dec 10, 2004

### Bernoulli

I must have been away when they told us that.

4. Dec 10, 2004

### 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.

5. Dec 10, 2004

### 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.

6. Dec 10, 2004

### 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.