Banach and Hilbert spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 18 May 2011

Banach and Hilbert spaces

Let be the field of complex numbers. A complex vector space is an abelian group V with a function ×VV such that

(a)   If c and v1, v2 V then c(v1+ v2) =cv1+ cv2 ,
(b)   If c1, c2 and v V then (c1+ c2)v =c1v+ c2v ,
(c)   If c1, c2 and v V then c1(c2v) = (c1c2)v ,
(d)   If vV then 1v=v.
Let X and Y be complex vector spaces. A linear transformation from X to Y is a function T:XY such that
if c1, c2 and x1, x2 X     then     T( c1x1 + c2x2 ) = c1T( x1) +c2T( x2) .
The morphisms in the category of vector spaces are linear transformations.

A topological vector space is a complex vector space V with a topology such that addition and scalar multiplication are continuous maps. The morphisms in the category of topological vector spaces are continuous linear transformations.

Let V be a complex vector space. A set CV is convex if C satisfies

if x,yC , t[0,1] then tx+ (1-t)y C .
A topological vector space V is locally convex if V has a basis of neighbourhoods of 0 consisting of convex sets.

Let V be a complex vector space and let C V be a convex subset of V. A function f:C is convex if f satisfies

if x,yC , t[0,1] then f(tx+ (1-t)y ) tf(x) + (1-t)f(y) .

HW: Show that the exponential function exp: (a,b) is convex.

A normed linear space is a complex vector space V with a function :V 0 such that

(a)   if x,yV then x+y x + y,
(b)   if c and vV then cv = |c| v ,
(c)   if vV and v =0 then v=0.

HW: Show that if V is a normed linear space then the map :V 0 is uniformly continuous, V is a metric space with respect to the metric d:V×V 0 defined by

d(x,y) = x-y
and, with the metric space topology, V is a topological vector space.

A Banach space is a normed linear space X such that X is a complete metric space with respect to the metric d:X×X 0 defined by

d(x,y) = x-y .
Let X and Y be normed linear spaces. An isometry from X to Y is a linear transformation T:XY such that
if xX     then     Tx = x .
Let X and Y be normed linear spaces. The norm of a linear transformation T:XY is
T = sup{ Tx | xX such that x 1 } .
A linear transformation is bounded if T <.

HW: If X and Y are normed linear spaces such that points are closed then a linear transformation T:XY is continuous iff it is bounded (reference???)

HW: Show that if X and Y are normed linear spaces then B(X,Y) ={bounded linear transformations φ:XY} with is a normed linear space and that if Y is a Banach space then B(X,Y) is a Banach space.

A Hilbert space is a complex vector space V with a function ,: V×V such that

(a)   if v1, v2V then v1, v2 = v2, v1 ,
(b)   if c1, c2 and v1, v2, v3V then c1 v1 + c2 v2 , v3 = c1 v1, v3 + c2 v2, v3 ,
(c)   if vV and v,v =0 then v=0,
(d)   V is a Banach space with norm :V0  given by      v2 =v,v.

Let V be a Hilbert space and let T:VV be a linear transformation. The adjoint of T is the linear transformation

T* :VV    defined by     T v1, v2 = v1 , T* v2 ,
for v1, v2V. The linear transformation T:VV is unitary if T satisfies
if v1, v2V     then     Tv1, Tv2 = v1 , v2 .

Duals

Let X with :V 0 be a normed linear space. Define

X* = {φ:X | φis a linear transformation and φ<},
where
φ = sup{ φ(x) | xX such that x 1 } .
Then, see [Ru, 5.21 and Ch. 5 Ex. 8],
(a)   X* is a Banach space.
(b)   X* separates points on X, i.e. if x1,x2 X and x1 x2 then there exists φ X* such that φ(x1) φ(x2) .
(c)   The map
ι: X X** x ιx: X* φ φ(x)
is an injective linear map such that ιx = x .
The construction of f in part (b) is a special case (or corollary) of the Hahn-Banach theorem, see [Ru, 5.21 and Theorem 5.20].

If M is a subspace of a normed linear space X and φ:M is a bounded linear functional then there exists a bounded linear functional Φ:X such that

(a)   if mM then Φ(m)= φ(m),
(b)   Φ = φ .

The proof of this theorem is essentially by induction, where the induction step extends φ from M to M+x0 for a vector x0 which is not in M.

Notes and References

These notes were synthesized from [Ru], [Kirillov], ... ????? They have evolved over the years through graduate courses in "Representation Theory" at University of Wisconsin, Madison and a course in "Measure Theory" at the Masters level at University of Melbourne. This presentation follows [Ru, Chapters 4 and 5].

References

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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