3) Show that ^2 + (b_1+b_2+...+b_n)^2})
for all real values of the variables. Furthermore, give a condition for equality to hold.
This is just the triangle inequality, as kamil said. I'm guessing you're using the Cauchy-Schwarz inequality in the form
If we let
)
and
)
, then this is equivalent to
where
is the dot product of
and
, and
is the magnitude of
respectively.
Now note that
, and hence
 \cdot (\mathbf{u}_1 + \mathbf{u}_2) \\<br />& = \mathbf{u}_1 \cdot \mathbf{u}_1 + \mathbf{u}_1 \cdot \mathbf{u}_2 + \mathbf{u}_2 \cdot \mathbf{u}_1 + \mathbf{u}_2 \cdot \mathbf{u}_2 \\<br />& = \|\mathbf{u}_1\|^2 + 2 \mathbf{u}_1 \cdot \mathbf{u}_2 + \|\mathbf{u}_2\|^2<br />\end{split})
as the dot product is linear and commutative. Then by the Cauchy-Schwarz inequality,
^2 \end{split})
and taking square roots yields the triangle inequality. By induction, this implies that
where
)
. It remains to note that
^2 + (b_1 + \ldots b_n)^2})
and that
Note that you can prove all this without writing it in the language of vectors, dot products, and magnitudes, but it's in some sense the most "natural" way. Also as kamil said, you don't need the Cauchy-Schwarz inequality to prove the triangle inequality, but it's normal to do so in that order, rather than the other way around.
The Cauchy-Schwarz inequality is a particular feature of a mathematical structure called an
inner product space, which is a
vector space with an inner product, a function that measures angles between vectors and magnitudes of vectors; the dot product on the vector space
is the standard example. Inner products in particular give rise to
norms, which is the magnitude of a vector, and norms satisfy the triangle inequality. This stuff is very important mathematically; the theory of
Banach spaces (
complete normed vector spaces) and
Hilbert spaces (complete inner product spaces) are the foundations of an area of mathematics called functional analysis, and have applications to many areas of mathematics and physics.
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