A metric linear space is a linear space equipped with metric but i want to know the point wise differences between metric space and metric linear space.Can any body write it down in points?
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A metric space can be any set, with or without an additional algebraic structure.
In a metric linear space $X$ there is a close connection between the distance and the linear algebraic structure:
The distance is translation invariant: $d(x+a,y+a)=d(x,y)$ for all $x,y,a\in X$. This implies that the sum is a continuous function from $X\times X$ to $X$.
The map $\mathbb{R}\times X\colon\to X$ that takes $(\lambda,x)$ to $\lambda\,x$ is continuous.
For instance $\mathbb{R}^3$ is a linear metric space with euclidean distance. The sphere $\{(x,y,z):x^2+y^2+z^2=1\}$ is also a metric space with the same distance, but is not a linear space.
Julián Aguirre
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AFAIK, "metric linear space" usually implies that the metric is translation invariant. – Giuseppe Negro Mar 07 '15 at 17:23
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Yes, I did not realize that. – Julián Aguirre Mar 07 '15 at 17:28
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can u tell me example of metric linear space. – Pushpa Shrestha Mar 23 '15 at 02:15
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Examples: 1) $\mathbb{R}^n$ with $d(x,y)=\bigl(\sqrt{x_1^p+\dots+x_n^p}\bigr)^{1/p}$, $1<p<\infty$. The case $p=2$ is the Euclidean metric. 2) The space $C([a,b])$ of continuous functions on a compact interval with the uniform distance $d(f,g)=\sup_{a\le x\le b}|f(x)|$. – Julián Aguirre Mar 23 '15 at 10:30