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Every linear function on a finite-dimensional space is continuous.

I was wondering what the domain and codomain of such linear function are?

Are they any two topological vector spaces (not necessarily the same), as along as the domain is finite-dimensional? Can the codomain be a different normed space (and may not be finite-dimensional)?

I asked this because I saw elsewhere the same statement except the domain is a finite-dimensional normed space, and am also not sure if the codomain can be a different normed space (and may not be finite-dimensional).

Thanks and regards!

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The codomain can we any normed space, finite dimensional or not. In fact, you can check that if $E$ is a vector space of dimension $n$ over the field of real numbers and $F$ any normed space, and $T\colon E\to F$ a linear map then $T(E)$ has a dimension $\leq n$. $T$ is continuous since we can take a basis $v_1,\ldots,v_k$ of $R(T)$ and we write for $x\in E$ $T(x)=\sum_{i=1}^ka_i(x)v_i$ and check that each $a_i$ is linear and continuous. –  Davide Giraudo Feb 24 '12 at 20:17
    
Continuity of a linear map is equivalent to boundedness: math.stackexchange.com/questions/86451/… –  dls Feb 24 '12 at 20:52
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To emphasize the key point: a "topological vector space" is more than a vector space endowed with a topology. It is a vector space endowed with a topology making the vector space operations continuous. When $V$ is finite dimensional over $\mathbb{R}$ or $\mathbb{C}$, then there is a unique "topological vector space" topology on $V$ (and any norm on $V$ induces it). This uniqueness means that people often talk about "finite dimensional spaces" without explicitly mentioning the topology (as that Wiki page does). But a specific topology is always intended. The result is not true in general. –  leslie townes Feb 24 '12 at 23:06
    
For example, there are lots of nonconstant linear functions from $\mathbb{R}$ with its usual vector space structure, to $\mathbb{R}$ with its usual vector space structure (these are the maps $x \mapsto ax$, with $a$ nonzero). But if you endow the domain with the usual topology, and the codomain with the discrete topology, then none of these nonconstant linear maps are continuous. So, to get the result you want, it's crucial that specific topologies are understood on the spaces (and indeed, the properties of these topologies are used heavily in any proof of that result). –  leslie townes Feb 24 '12 at 23:09
    
@leslietownes: Thanks! "When V is finite dimensional over R or C, then there is a unique "topological vector space" topology on V (and any norm on V induces it)." Do you mean that V is firstly a vector space, and then you endow V with a topology from its base field R or C? If yes, how do you do that? –  Tim Feb 25 '12 at 1:23

2 Answers 2

up vote 12 down vote accepted

The result we can show is the following:

Let $E$ and $F$ two topological vector spaces, and $T\colon E\to F$ a linear map. If $E$ is finite dimensional, then $T$ is continuous.

First, if $(e_1,\ldots,e_n)$ is a basis of $E$, then any set of $n+1$ vectors of $T(E)$ is linearly dependent, so $T(E)$ has a dimension $\leq n+1$. Let $k$ be the dimension of $T(E)$, and $(v_1,\ldots,v_k)$ a basis of this space. We can write for any $x\in E$: $T(x)=\sum_{i=1}^ka_i(x)v_i$ and since $v_i$ is a basis each $a_i$ is linear. We have to show that each map $T_i\colon E\to F$, $T_i(x)=:a_i(x)v_i$ is continuous. By definition of a topology on a topological vector space we only have to show that the map $x\mapsto a_i(x)$ is continuous. To do that, we use the fact that a finite dimensional topological vector space can be equipped with a norm which gives the same topology (in fact it's the unique one), namely put $$N(\sum_{j=1}^n\alpha_jx_j):=\sum_{j=1}^n|\alpha_j|.$$ Now the continuity is easy to check: denoting $x=\sum_{j=1}^nx_je_j$ and $y=\sum_{j=1}^ny_je_j$ $$|a_i(x)-a_i(y)|\leq \sum_{j=1}^n|a_i((x_j-y_j)e_j)|=\sum_{j=1}^n|x_j-y_j|\cdot |a_i(e_j)|\leq N(x-y)\sum_{j=1}^n|a_i(e_j),$$ since $|x_j-y_j|\leq N(x-y)$ for all $1\leq j\leq n$.

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Thanks! (1) "a finite dimensional topological space can be equipped with a norm which gives the same topology", do you mean topological vector space instead of topological space? (2) Why is the last inequality? –  Tim Feb 25 '12 at 1:06
    
Yes I meant topological vector space, I will fix it and give more details for the last inequality. –  Davide Giraudo Feb 25 '12 at 8:49
    
Thanks! (1) In the definition of norm $N$, I suppose $x_j$ is $e_j$? (2) Regarding your detail explanation about the equality asked in my last comment, is there an inequality for a general normed space, like $||x .\times y|| \leq ||x|| \cdot ||y||$ for some elementwise multiplication $.\times$ between two vectors? In order to make "element" to have sense, do we have to make $.\times$ independent of basis? –  Tim Feb 25 '12 at 12:54
    
Isn't $T(E)$ of dimension $\le n$? (assuming $E$ is of dimension $n$) –  Matt N. Jun 14 at 13:05

There is a nice write-up of a proof of a simpler version of this (for normed vector spaces) on Wikipedia. Let me reproduce the proof given on Wikipedia:

If $T:X\to Y$ is a linear map where $X,Y$ are normed vector spaces and $X$ has finite dimension then $T$ is continuous.

Proof: Let $e_1,\dots, e_n$ be a basis of $X$. Then for $x \in X$ we have

$$ \|Tx\|_Y = \|T \sum_i \alpha_i e_i\|_Y = \|\sum_i \alpha_i Te_i \|_Y \le \sum_i |\alpha_i|\|Te_i\|_Y$$

Let $\varepsilon >0$ be given and let $M = \max_i \|Te_i\|_Y$. Let $\delta = {\varepsilon \over M}$. Then for $x$ with $\|x\|,\delta$ we have

$$ \|Tx\|_Y \le \sum_i |\alpha_i|\|Te_i\|_Y \le M \sum_i |\alpha_i| < \varepsilon$$

hence $T$ is continuous.

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