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So if $X$ is a vector space, and you define a norm, $x \mapsto \| x \|$, on it, then the bounded subset, $V = \{ x \in X: \|x\| < \infty \}$ is automatically a subspace. This follows from the definition of a norm, so for all $x,y \in V, c \in \mathbb{C}$, $\|x + c y\| \le \|x\| + \|cy\| = \|x\|+ |c| \|y\| < \infty$, so $x+cy \in V$.

Is this the reasoning behind the definition of a norm?


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In a normed vector space every vector has finite norm. – Rasmus Jan 10 '11 at 7:47
Note that for your definition $V=X$. – mpiktas Jan 10 '11 at 8:14

The short version is that we use norms to help us study convergence in function spaces; this is the huge subject of functional analysis, which has applications everywhere in mathematics (PDEs, mathematical physics, probability, representation theory...). I recommend Dieudonne's History of Functional Analysis if you are really interested in the full story.

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This might be helpful.

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