I am an undergraduate student and we have both linear algebra and metric space course in this semester.In linear algebra we studied norm and in metric spaces we studied metric.Now studying some examples I think every metric on vector space may define a norm and every norm can define a metric.I am sure about the last one but does every metric on vector space give rise to a norm?
I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function  $d(u,v) = \lVert u - v \rVert$, $u,v \in V$.   My question is whether every metric on a linear space can be induced by norm? I know answer is  no but I need proper justification.   Edit: Is there any method to check whether a given metric space is induced by norm ?  Thanks for help
I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function  $d(u,v) = \lVert u - v \rVert$, $u,v \in V$.   My question is whether every metric on a linear space can be induced by norm? I know answer is  no but I need proper justification.   Edit: Is there any method to check whether a given metric space is induced by norm ?  Thanks for help