# Thread: Vector Subspaces

1. ## Re: Vector Subspaces

Originally Posted by Plato
That is exactly what I am saying.
Zero is a finite number.
You are confusing the difference between I have two books and I have 2 books.

Two books: "War and Peace," "Uncle Vanya."
2 books: 2, 2, 2, 2, 2,

2. ## Re: Vector Subspaces

A set that has no entries is empty. A set that has zero entries is finite. A set that has a zero number of entries is empty.

3. ## Re: Vector Subspaces

Originally Posted by Hartlw
A set that has no entries is empty. A set that has zero entries is finite. A set that has a zero number of entries is empty.
Imprecisions of usual language. What about the following?:

Definition We say that the sequence $\displaystyle (x_n)_{n\geq 0}$ has finitely many non zero terms iff there exists $\displaystyle p\in\mathbb{N}$ such that $\displaystyle x_n=0$ for all $\displaystyle n\geq p$ .

This is just what all mathematicians interpret with the expression the sequence $\displaystyle (x_n)_{n\geq 0}$ has finitely many non zero terms.

Question: Has the zero sequence finitely many non zero elements?. You are going to say yes.

4. ## Re: Vector Subspaces

Originally Posted by FernandoRevilla
Imprecisions of usual language. What about the following?:

Definition We say that the sequence $\displaystyle (x_n)_{n\geq 0}$ has finitely many non zero terms iff there exists $\displaystyle p\in\mathbb{N}$ such that $\displaystyle x_n=0$ for all $\displaystyle n\geq p$ .

This is just what all mathematicians interpret with the expression the sequence $\displaystyle (x_n)_{n\geq 0}$ has finitely many non zero terms.

Question: Has the zero sequence finitely many non zero elements?. You are going to say yes.

I have a sequence which has (only) three non-zero terms. Can the sequence be 0? No

I have a sequence which has no terms* which aren't zero. The sequence could be 0, but nothing else.

I have a sequence which has only no terms which aren't zero. The sequence has to be zero.

So W is a subspace because it contains only 0.

* a zero number of non-zero terms. zero is a finite number. Reference Plato

5. ## Re: Vector Subspaces

Originally Posted by Hartlw
I have a sequence which has (only) three non-zero terms. Can the sequence be 0? No

I have a sequence which has no terms* which aren't zero. The sequence could be 0, but nothing else.

I have a sequence which has only no terms which aren't zero. The sequence has to be zero.

So W is a subspace because it contains only 0.

* a zero number of non-zero terms. zero is a finite number. Reference Plato
Actually, if finite means nore than a zero number of terms, the answer is W is not a subspace because it doesn't contain 0.
If "finite number" is restricted to "zero number," W is a subspace because it only contains 0

6. ## Re: Vector Subspaces

Originally Posted by Hartlw
This is what I am saying: A sequence of zeros does not have any non-zero entries, so it cannot have finiteley many of them. 0 does not belong to W.

If I have no books do I have a finite numbr of them?

What are you saying?
You won't have a chance to reply to this as I have closed the thread but consider your definition of a finite set, then check how a selection of text or reference books define a finite set.

CB

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