Question

Please show that the space of bounded functions on a set X, l^∞(X)={f∈R^X |There is M∈R so that for all x∈X, we have |f(x)|≤M} is a subspace of R^x. Or give me a hint on this problem? Thanks

Answer 1

So, prove that:
\[
I^\infty(X)=\{f\in\mathbb{R}^X|\exists M\in\mathbb{R} \text{ s.t. }\forall x\in X, |f(x)|\le M\}
\]Is a subspace of \(\mathbb{R}^x\)? (Just to make sure I'm understanding the problem, correctly.

And, what do you mean when you say \(f\in \mathbb{R}^X\)? \(X\) is a set, do you mean its cardinality?

Answer 2

I'll try to rewrite the question...

"Show that on a set X, the space of bounded functions \[l^∞(X) = \left\{ f \in \mathbb{R}^X | \exists M \in \mathbb{R}; s.t. \forall x \in X, |f(x)|≤M \right\},\] is a subspace of \[\mathbb{R}^X.\] In terms of what I mean when I say: \[f \in \mathbb{R}^X\] ... well that's just what it says in this homework question and the whole question is confusing me, so I'm not going to be of very much help, unfortunately. Sorry!

Answer 3

Screen shot of the question...

Answer 4

Hmm... I think I get what your professor means by \(f\in\mathbb{R}^X\). I think it means:
\[
f:X\mapsto \mathbb{R}
\]It's a little odd he or she didn't define that in class or on the homework...
All right, so, we know that:
\[
l^\infty(X)\subset\mathbb{R}^X
\]By definition. Now, let's declare two functions \(f, g \in l^\infty(X)\). We know that, for such to be a subspace of \(\mathbb{R^X}\), \(f+g\in l^\infty(X)\) and, \(\forall c\in\mathbb{R}\), \(cf\in l^\infty(X)\). Let's prove these statements:
If:
\[
f, g \in l^\infty(X)\implies\\
\exists M_f\in \mathbb{R}, s.t., \forall x\in X, f(x)\le M
\]We declare the same for \(M_g\), for function \(g\). Then:
\[
f+g\le M_f+M_g
\]Since:
\[
M_f+M_g\in\mathbb{R} \wedge f+g\in R^X\implies f+g\in l^\infty(X)
\]And, we do the same, for some \(f\in l^\infty(X)\), \(c\in\mathbb{R}\):
\[
\forall x, |f(x)|\le M\implies\\
c|f(x)|\le cM
\]And, since:
\[
c\cdot f(x)\in\mathbb{R}^X
\]We are done, thus, \(l^\infty(X)\) is a subspace of \(\mathbb{R}^X\)

Answer 5

Sorry, I forgot to put absolute values around some things, but I'm sure you get what I mean.

Answer 6

Oh, and, for the last part:
\[
cM\in\mathbb{R}
\]