Equality of two elements of a hilbert space defined?

[pellman]

Given x,y elements of a hilbert space H, how do we conclude that x = y? Since there is an inner product, we can say that x = y only if (x,z) = (y,z) for all z in H. But is there a definition of equality which does not depend on the inner product?

A hilbert space is a special instance of vector space. According to wikipedia https://en.wikipedia.org/wiki/Vector_space#Definition the definition of a vector space is given in terms of properties relating elements of the space that depend on a prior understanding of equality, e.g., commutativity with respect to addition. What does equals mean in this case?

 

[Dr. Courtney]

If two vectors are equal, all of their components are equal in how every many dimensions the vectors are defined.

If two wave functions are equal, the magnitudes of each component must also be equal in any basis that spans the Hilbert space.

 

[FactChecker]

If x-y=0, then x=y. Equality of elements must be defined for any set. Equality in a Hilbert Space can be proven by any of the methods that it inherits. For instance, a Hilbert space is an abstract group. For an abstract group, proving that x-y=0 would prove that x=y. So x-y=0 implies x=y in any Hilbert space. As Dr. Courtney's method shows, the proofs for a vector space will also work in a Hilbert space because all Hilbert spaces are vector spaces.

 

[pellman]

If x-y=0, then x=y. Equality of elements must be defined for any set. Equality in a Hilbert Space can be proven by any of the methods that it inherits. For instance, a Hilbert space is an abstract group. For an abstract group, proving that x-y=0 would prove that x=y. So x-y=0 implies x=y in any Hilbert space. As Dr. Courtney's method shows, the proofs for a vector space will also work in a Hilbert space because all Hilbert spaces are vector spaces.

But the definition of the zero vector requires a prior concept of equality. The zero vector 0 is such that 0 + x = x for all x.

 

[pellman]

If two vectors are equal, all of their components are equal in how every many dimensions the vectors are defined.

If two wave functions are equal, the magnitudes of each component must also be equal in any basis that spans the Hilbert space.

I think my suggestion about using the inner product is probably equivalent to this.

 

[Dr. Courtney]

I think my suggestion about using the inner product is probably equivalent to this.

Perhaps, but you did not specify how you intended to use the inner product, you asked if there was a way to define equality without using the inner product. Consider ordinary vectors in 3D Cartesian space. If vector A = vector B, then

A_x = B_x,
A_y = B_y, and
A_z = B_z

This is the real meaning of equality among vectors. One might define or describe an equivalent test for equality using the dot product, but that does not mean that one cannot define equality without using the inner product.

 

[Fredrik]

What does equals mean in this case?

Same thing as it does for real numbers. When x and y are symbols that represent elements of some set S, the string of text "x=y" is meant to be interpreted as the statement "the symbols x and y represent the same element of S". So to ask what equality means in the context of Hilbert spaces is to ask what "the same" means in plain English.

Equality of vectors in a Hilbert space isn't defined using the inner product any more than equality of real numbers is defined using multiplication. The set of real numbers is a Hilbert space by the way. (A real Hilbert space, not a complex one). The usual multiplication operation is the inner product.

It makes more sense to ask what equality means when the particular Hilbert space you're dealing with is defined in some complicated way like "H is the set of equivalence classes of Cauchy sequences in the inner product space K". (That's not a complete definition. Before you say this, you have to define the equivalence relation, and after you've said it, you have to define addition, scalar multiplication and the inner product). If x has been defined as the equivalence class that contains the Cauchy sequence S and y has been defined as the equivalence class that contains the Cauchy sequence T, then you would have to use the definition of the equivalence relation to check if S is equivalent to T. But if your Hilbert space is an arbitrary Hilbert space, because you're trying to prove a theorem that's supposed to hold for all Hilbert spaces, then equality isn't defined from some other mathematical concept.

"only if" means ⇒, not ⇐.

 

[pellman]

Same thing as it does for real numbers. When x and y are symbols that represent elements of some set S, the string of text "x=y" is meant to be interpreted as the statement "the symbols x and y represent the same element of S".

Ok. I was coming at this from the notion that we can only show that x and y are equal if there is some property that "equal" means and then showing that this property is true of x and y. It could not be "equals means that x - y = 0" because the definition of the 0 itself requires the concept of equals.

Instead now I see it like this: equals is a fundamental concept which means "these symbols stand for the same thing". Then we define the 0 element in terms of the addition operation as 0 + x =x for all x, where here again we mean by equals " the symbols on each side of the equal sign represent the same number or vector."

Then if we have two symbols representing numbers (or vectors) x and y, and we are wondering, "Are they equal?", we can conclude they are equal if we can show that x - y = 0. But "x = y is equivalent to x - y = 0" is a property of the zero element, not a fundamental definition of equality.

I think I am satisfied with this ... though I still have a tiny doubt that there isn't a circularity hiding in this.

 

[FactChecker]

Instead now I see it like this: equals is a fundamental concept which means "these symbols stand for the same thing".

Exactly. Defining what '=' means is done very early in defining a set. That is before any operations, vector space, or inner product are defined. The definition of '=' often sets up equivalence classes. You may have a set of distinct elements where you only want to consider certain aspects and ignore all others. Then you set up equivalence classes where two elements will be considered equal if the properties you care about are equal even if their ignored properties are different.

Example: Among all people, you may want to consider only the gender and age of people while ignoring everything else. Then any two men of equal age are equal, any two women of equal age are equal, no man equals a woman, and no people of different age are equal. That would define a set whose elements can be indicated by (gender, age). Each element is an equivalence class of people of the same gender and age.

 

[Fredrik]

The axiom that appears to define = in ZFC set theory is a funny thing. It says that for all E and all F, if E and F have the same elements, then E=F. This is an implication rather than an equivalence because the converse (if E=F, then E and F have the same elements) is considered a consequence of the meaning of the equality symbol, which apparently is viewed as even more fundamental than the ZFC axioms.

I would say that the axiom gives the = symbol a second meaning, in addition to the one it already had.

 

[micromass]

The axiom that appears to define = in ZFC set theory is a funny thing. It says that for all E and all F, if E and F have the same elements, then E=F. This is an implication rather than an equivalence because the converse (if E=F, then E and F have the same elements) is considered a consequence of the meaning of the equality symbol, which apparently is viewed as even more fundamental than the ZFC axioms.

I would say that the axiom gives the = symbol a second meaning, in addition to the one it already had.

Hmm. I guess there are two approaches. One approach is to use the extensionality axiom to define =. In this case, you don't have a previous equality symbol, and you see the equality relation $x=y$ as a shorthand for $\forall a: a\in x\Leftrightarrow a\in y$.

In the other approach, both = and $\in$ are undefined notions. The extensionality axiom then merely states a relationship between = and $\in$, and this cannot be seen as a definition of = and $\in$.

 

[FactChecker]

The posts are migrating to a discussion of equality of sets. The original post was asking about equality of two elements, which is more basic.