F**S
发帖数: 13 | 1 I am embarrassed to ask such elementary questions but they have bothered me
like a bone in my throat for quite a while. Please be so kind to enlighten
me...
The questions are centered closely about infinite dimensionality:
How exactly to see that a mapping space is infinite dimensional? To be
specific, take the simplest example of the vector space, C(R, R), of
continuous functions from R to R, where R stands for the space of real
numbers. The statement must be obvious since books I have consulted simply
give the assertion. However, I seem to have a hard time pin-point the
argument...
Moreover, how to find, if possible, an explicit (Hamel) basis for this
infinite-dimensional space C(R, R)? Here I mean in the exclusive sense of
linear algebra, ignoring any topological structures that may be naturally
endowed thereon.
Lastly, am I correct in thinking that the cardinality of the dimension for a
function space, be it countably or uncountably infinite, cannot be
determined without possibly imposing an appropriate norm structure with
respect to which the space is complete? In other words, insofar as purely
linear-algebraic considerations are taken into account, the best one can say
is that the space is infinite-dimensional, not any refinement on (un)
countability...
Any insight will be greatly appreciated!:) | B****n
发帖数: 11290 | 2 C(R,R)的basis問題: 必然牽扯到一點分析 不可能純粹在linear algebra裡面討論 因
為牽扯到了極限(任何一連續函數 可被basis裡的函數的linear combination所逼近)
任何一本高微的課本應該都有 不需要用到什麼topology
C(R,R)是infinite dimensional的問題: 你只需要證明span{x, x^2,...,x^n}是n維的
for any n
me
【在 F**S 的大作中提到】
: I am embarrassed to ask such elementary questions but they have bothered me
: like a bone in my throat for quite a while. Please be so kind to enlighten
: me...
: The questions are centered closely about infinite dimensionality:
: How exactly to see that a mapping space is infinite dimensional? To be
: specific, take the simplest example of the vector space, C(R, R), of
: continuous functions from R to R, where R stands for the space of real
: numbers. The statement must be obvious since books I have consulted simply
: give the assertion. However, I seem to have a hard time pin-point the
: argument...
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