A lot depends on why you don't want to write to another file, but here are some options:

1) (assuming you have very limited disk-space, but enough memory to read in file)

- read the file in, remove spaces in memory then write it back

open my $fh, 'uuu.txt';
my @new = map { s/\s+//g; } <$fh>;
close $fh;
open $fh '>uuu.txt';
print $fh @new;
close $fh;
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2) (not enough memory but say 20% margin of temporary disk space)

- for each line read in, remove spaces and write to a compressed file, perhaps on /tmp if it has the space.

- uncompress new file over old

- remove compressed file

open my $fh, 'uuu.txt';
open my $gh, '| bzip2 -9 -c > /tmp/uuu.txt.bz2'; # or gzip if no bzip2
while (<$fh>) {
    s/\s+//g;
    print $gh $_ or suffer();
}
close $fh;
close $gh or suffer();
system 'bunzip2 -c /tmp/uuu.txt.bz2 > uuu.txt';
unlink '/tmp/uuu.txt.bz2';

sub suffer {
   warn "$!: /tmp/uuu.txt.bz2\n"; # e.g. $! reports disk full
   unlink '/tmp/uuu.txt.bz2';
   exit 256;
}
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Although if there is no such resource problem, better say what the reason is!