Question

What's the paper of Schwarz inequality and triangle inequality?

Answer 1

Do you mean the proof of the Schwarz inequality and triangle inequality?

Assuming that's what you want, then this is certainly my favorite ( attributed, I believe, to John von Neumann ):

The norm/length of any vector is positive. Let t be any real number and any two vectors V and W be given. Then examine the vector V - tW :
\[|V - tW|^{2} \ge 0\]\[(V -tW, V- tW)\ge0\][ where (A,B) is the inner product of the vectors A and B, and also relying on homogeneity, symmetry and additivity of said inner product ]
\[(V, V) -2t(V,W) + t^{2}(W,W)\ge0\]\[|V|^{2} - 2t(V,W) +t^{2}|W|^{2}\ge0\]Yielding a quadratic expression in t, which is always non-negative. This requires a non-positive discriminant, in this instance with
\[a = |W|^{2}\]\[b = -2(V,W)\]\[c = |V|^{2}\]then
\[b^{2} - 4ac \le0\]becomes
\[4(V,W)^{2} - 4|W|^{2}|V|^{2}\le0\]or
\[(V,W)^{2} \le |W|^{2}|V|^{2}\]then taking square root of both sides
\[|(V,W)|\le|V||W|\]I think this is an amazingly sneaky proof, dirty tricks, or if you like : brilliantly elegant! :-)
You can then get the triangle inequality from Cauchy-Schwarz via:
\[|V+W|^{2} = (V+W,V+W)\]\[= (V,V)+2(V,W)+(W,W)\]\[\le(V,V)+2|(V,W)|+(W,W)\]now use Cauchy-Schwarz
\[\le(V,V)+2|V||W|+(W,W))\]\[=|V|^{2}+2|V||W|+|W|^{2}=(|V+|W|)^{2}\]and now take the square root of both sides to get:
\[|V+W|\le|V|+|W|\]That's one form of the triangle inequality at least, others are derivable from that.

Answer 2

Wonderful proof!!!! This Schwarz inequality is used for what??

Answer 3

Pretty well anywhere there is vector-like behaviour ie. vector spaces. The interesting part is : what entities behave like vectors? The list is surprisingly long, and often not immediately apparent. Look for areas that have linear combinations of things, orthogonality of things and such. Don't constrain yourself to thinking in purely geometric terms. Use analogy.

Quantum mechanics, for instance, assumes an "L2 Hilbert space of functions" where the dynamical state of a system is represented by 'wave functions'. These are even called 'state vectors'. Linearity of such vectors means that one can add combinations of said functions, with suitable but now complex number multipliers, and get another function which represents yet another valid dynamical state of the system. Very much like adding geometric vectors. Orthogonality here implies that if you are definitely in one type of state, usually called a basis state, then you can't be in another. So that's like geometric vectors being at right angles to each other : no part of one vector has a component lying in the direction of the other vector. The general idea is to discover/solve the right differential equation to describe the change of the system from one state to another as time passes, which is rather like an ordinary position vector changing to track a moving object in 3D space under some rules of motion. One may then apply 'operators' to these functions/vectors to extract information ( 'observables' ) in a manner reminiscent of 'how long is this vector' or 'what is this vector's components in such and such direction' ?

Hilbert refers to one of the guys that defined much of this about a century ago. The 'L' in L2 refers to Lebesque who discovered generalisations of the Riemann integral ( see 'measure theory' ), where L2 specifically implies a certain restraint ( 'square integrable' which in physics means of 'finite power' - corresponding to finite length in vector terminology ) on the functions that can be acceptable for use.

Answer 4

Sorry, 'Lebesgue' - I always get that wrong. :-)