If zero element doesn't exist in a vector space,then does that mean that additive inverse can also not exist? For example, take a set V of all real polynomials with degree 4 or 6 with the usual scalar multiplication and vector addition, then is V a vector space? If not, then what properties does it not satisfy?
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1$\begingroup$ A real vector space $V,+,\cdot$ is a commutative group $V,+$ and there is a scalar multiplication $\cdot:\mathbb{R}\times V\rightarrow V$ such that $1\cdot v=v$ and it satisfies a left and right distributive identity. The fact that it is a commutative group implies that it must have a neutral element. Thus the space of polynomials of degree 4 and 6 cannot be a vector space as the neutral element is the zero polynomial (whose degree is -1 by convention). Vector spaces in general, do not carry a notion of inverses other than inverses w.r.t. the group structure. $\endgroup$– Mathematician 42Feb 10, 2018 at 16:00
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$\begingroup$ So,can we say that since there is no zero element in the set,additive inverse can not exist? $\endgroup$– IronVenomFeb 10, 2018 at 16:03
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$\begingroup$ If there were an element $v$ whose additive inverse $-v$ also belonged to the vector space, then $v+(-v)=1\cdot v+(-1)\cdot v=(1+(-1))\cdot v=0\cdot v=0$ should also belong to the vector space. $\endgroup$– Mathematician 42Feb 10, 2018 at 16:06
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2$\begingroup$ You can't even define the notion of "additive inverse" without mentioning a zero element, so you can't begin to ask whether additive inverses exist when you don't have a zero. $\endgroup$– Ethan BolkerFeb 10, 2018 at 16:06
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$\begingroup$ @Mathematician42: I suggest you answer the question using the "answer" box and not the "comment" box. $\endgroup$– Martin ArgeramiFeb 10, 2018 at 16:37
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A real vector space $V,+,\cdot$ is a commutative group $V,+$ and there is a scalar multiplication $\cdot:\mathbb{R}\times V\rightarrow V $such that $1⋅v=v$ and it satisfies a left and right distributive identity. The fact that it is a commutative group implies that it must have a neutral element. Thus the space of polynomials of degree 4 and 6 cannot be a vector space as the neutral element is the zero polynomial (whose degree is -1 by convention). Vector spaces in general, do not carry a notion of inverses other than inverses w.r.t. the group structure.
As indicated in the comments, one cannot speak about inverses without the notion of a neutral element first.
answered Feb 10, 2018 at 18:04