<s>
Riesz	B-Algorithm
's	I-Algorithm
lemma	I-Algorithm
(	O
after	O
Frigyes	O
Riesz	O
)	O
is	O
a	O
lemma	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
The	O
lemma	O
may	O
also	O
be	O
called	O
the	O
Riesz	B-Algorithm
lemma	I-Algorithm
or	O
Riesz	O
inequality	O
.	O
</s>
<s>
The	O
proof	O
can	O
be	O
found	O
in	O
functional	B-Application
analysis	I-Application
texts	O
such	O
as	O
Kreyszig	O
.	O
</s>
<s>
Using	O
this	O
new	O
terminology	O
,	O
Riesz	B-Algorithm
's	I-Algorithm
lemma	I-Algorithm
may	O
also	O
be	O
restated	O
in	O
plain	O
English	O
as	O
:	O
</s>
<s>
When	O
is	O
trivial	O
then	O
it	O
has	O
no	O
vector	O
subspace	O
and	O
so	O
Riesz	B-Algorithm
's	I-Algorithm
lemma	I-Algorithm
holds	O
vacuously	O
for	O
all	O
real	O
numbers	O
The	O
remainder	O
of	O
this	O
section	O
will	O
assume	O
that	O
which	O
guarantees	O
that	O
a	O
unit	O
vector	O
exists	O
.	O
</s>
<s>
Riesz	B-Algorithm
's	I-Algorithm
lemma	I-Algorithm
is	O
always	O
false	O
when	O
because	O
for	O
every	O
unit	O
vector	O
the	O
required	O
inequality	O
fails	O
to	O
hold	O
for	O
(	O
since	O
)	O
.	O
</s>
<s>
However	O
,	O
every	O
finite	O
dimensional	O
normed	O
space	O
is	O
a	O
reflexive	O
Banach	O
space	O
,	O
so	O
Riesz	B-Algorithm
's	I-Algorithm
lemma	I-Algorithm
does	O
holds	O
for	O
when	O
the	O
normed	O
space	O
is	O
finite-dimensional	O
,	O
as	O
will	O
now	O
be	O
shown	O
.	O
</s>
<s>
Riesz	B-Algorithm
's	I-Algorithm
lemma	I-Algorithm
guarantees	O
that	O
for	O
any	O
given	O
every	O
infinite-dimensional	O
normed	O
space	O
contains	O
a	O
sequence	O
of	O
(	O
distinct	O
)	O
unit	O
vectors	O
satisfying	O
for	O
or	O
stated	O
in	O
plain	O
English	O
,	O
these	O
vectors	O
are	O
all	O
separated	O
from	O
each	O
other	O
by	O
a	O
distance	O
of	O
more	O
than	O
while	O
simultaneously	O
also	O
all	O
lying	O
on	O
the	O
unit	O
sphere	O
.	O
</s>
<s>
This	O
sequence	O
can	O
be	O
constructed	O
by	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
Riesz	B-Algorithm
's	I-Algorithm
lemma	I-Algorithm
can	O
be	O
applied	O
directly	O
to	O
show	O
that	O
the	O
unit	O
ball	O
of	O
an	O
infinite-dimensional	O
normed	O
space	O
is	O
never	O
compact	O
.	O
</s>
<s>
More	O
generally	O
,	O
if	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
locally	O
compact	O
,	O
then	O
it	O
is	O
finite	O
dimensional	O
.	O
</s>
<s>
Namely	O
,	O
if	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
finite	O
dimensional	O
,	O
it	O
is	O
locally	O
compact	O
.	O
</s>
<s>
But	O
compact	O
sets	O
are	O
bounded	B-Algorithm
,	O
so	O
lies	O
in	O
the	O
closure	O
of	O
This	O
proves	O
the	O
result	O
.	O
</s>
<s>
The	O
spectral	B-Algorithm
properties	I-Algorithm
of	I-Algorithm
compact	I-Algorithm
operators	I-Algorithm
acting	O
on	O
a	O
Banach	O
space	O
are	O
similar	O
to	O
those	O
of	O
matrices	O
.	O
</s>
<s>
Riesz	B-Algorithm
's	I-Algorithm
lemma	I-Algorithm
is	O
essential	O
in	O
establishing	O
this	O
fact	O
.	O
</s>
<s>
Riesz	B-Algorithm
's	I-Algorithm
lemma	I-Algorithm
also	O
shows	O
that	O
the	O
identity	O
operator	O
on	O
a	O
Banach	O
space	O
is	O
compact	O
if	O
and	O
only	O
if	O
is	O
finite-dimensional	O
.	O
</s>
