When does an inner product induce a norm?

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When we consider a vector space $V$ over some field $F$, I know that when the $F=mathbbR$ or $ =mathbbC$, by setting $|x|=leftlanglex,xrightrangle^frac12$ we get a norm. However, since the inner product is a function with its image in $F$, what happens if we consider any $V$ over the rational numbers? For example, if we take $mathbbQ^2$ over $mathbbQ$ with the dot product, then $v=(1,1)$ has norm $sqrt2$, which is not rational. How can one obtain a norm from a given inner product in such cases?










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    When we consider a vector space $V$ over some field $F$, I know that when the $F=mathbbR$ or $ =mathbbC$, by setting $|x|=leftlanglex,xrightrangle^frac12$ we get a norm. However, since the inner product is a function with its image in $F$, what happens if we consider any $V$ over the rational numbers? For example, if we take $mathbbQ^2$ over $mathbbQ$ with the dot product, then $v=(1,1)$ has norm $sqrt2$, which is not rational. How can one obtain a norm from a given inner product in such cases?










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      up vote
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      down vote

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      up vote
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      down vote

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      When we consider a vector space $V$ over some field $F$, I know that when the $F=mathbbR$ or $ =mathbbC$, by setting $|x|=leftlanglex,xrightrangle^frac12$ we get a norm. However, since the inner product is a function with its image in $F$, what happens if we consider any $V$ over the rational numbers? For example, if we take $mathbbQ^2$ over $mathbbQ$ with the dot product, then $v=(1,1)$ has norm $sqrt2$, which is not rational. How can one obtain a norm from a given inner product in such cases?










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      javierochomil is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      When we consider a vector space $V$ over some field $F$, I know that when the $F=mathbbR$ or $ =mathbbC$, by setting $|x|=leftlanglex,xrightrangle^frac12$ we get a norm. However, since the inner product is a function with its image in $F$, what happens if we consider any $V$ over the rational numbers? For example, if we take $mathbbQ^2$ over $mathbbQ$ with the dot product, then $v=(1,1)$ has norm $sqrt2$, which is not rational. How can one obtain a norm from a given inner product in such cases?







      linear-algebra normed-spaces inner-product-space






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      edited 1 hour ago









      zipirovich

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          An ordered field where $a^2+b^2$ is always a square is called a Pythagorean field. As you observe, not every ordered field is Pythagorean, but each
          ordered field has a Pythagorean extension. If you really want $L^2$-norms
          you could always extend your ground field to a Pythagorean extension field.






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          • Thanks, I didn't know about this extensions. It's not that I want a norm like the one I mention, but instead I would like to know how can one obtain a normed space from an inner product space when the field is, like you say, not Pythagorean.
            – javierochomil
            1 hour ago










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          1 Answer
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          1 Answer
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          up vote
          4
          down vote













          An ordered field where $a^2+b^2$ is always a square is called a Pythagorean field. As you observe, not every ordered field is Pythagorean, but each
          ordered field has a Pythagorean extension. If you really want $L^2$-norms
          you could always extend your ground field to a Pythagorean extension field.






          share|cite|improve this answer




















          • Thanks, I didn't know about this extensions. It's not that I want a norm like the one I mention, but instead I would like to know how can one obtain a normed space from an inner product space when the field is, like you say, not Pythagorean.
            – javierochomil
            1 hour ago














          up vote
          4
          down vote













          An ordered field where $a^2+b^2$ is always a square is called a Pythagorean field. As you observe, not every ordered field is Pythagorean, but each
          ordered field has a Pythagorean extension. If you really want $L^2$-norms
          you could always extend your ground field to a Pythagorean extension field.






          share|cite|improve this answer




















          • Thanks, I didn't know about this extensions. It's not that I want a norm like the one I mention, but instead I would like to know how can one obtain a normed space from an inner product space when the field is, like you say, not Pythagorean.
            – javierochomil
            1 hour ago












          up vote
          4
          down vote










          up vote
          4
          down vote









          An ordered field where $a^2+b^2$ is always a square is called a Pythagorean field. As you observe, not every ordered field is Pythagorean, but each
          ordered field has a Pythagorean extension. If you really want $L^2$-norms
          you could always extend your ground field to a Pythagorean extension field.






          share|cite|improve this answer












          An ordered field where $a^2+b^2$ is always a square is called a Pythagorean field. As you observe, not every ordered field is Pythagorean, but each
          ordered field has a Pythagorean extension. If you really want $L^2$-norms
          you could always extend your ground field to a Pythagorean extension field.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 1 hour ago









          Lord Shark the Unknown

          92.5k956120




          92.5k956120











          • Thanks, I didn't know about this extensions. It's not that I want a norm like the one I mention, but instead I would like to know how can one obtain a normed space from an inner product space when the field is, like you say, not Pythagorean.
            – javierochomil
            1 hour ago
















          • Thanks, I didn't know about this extensions. It's not that I want a norm like the one I mention, but instead I would like to know how can one obtain a normed space from an inner product space when the field is, like you say, not Pythagorean.
            – javierochomil
            1 hour ago















          Thanks, I didn't know about this extensions. It's not that I want a norm like the one I mention, but instead I would like to know how can one obtain a normed space from an inner product space when the field is, like you say, not Pythagorean.
          – javierochomil
          1 hour ago




          Thanks, I didn't know about this extensions. It's not that I want a norm like the one I mention, but instead I would like to know how can one obtain a normed space from an inner product space when the field is, like you say, not Pythagorean.
          – javierochomil
          1 hour ago










          javierochomil is a new contributor. Be nice, and check out our Code of Conduct.









           

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