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Course by: Mark A. Davenport. E-mail the author

Complete Vector Spaces

Module by: Mark A. Davenport. E-mail the author

Definition1

A complete normed vector space is called a Banach space.

Example 1

• ·C[a,b]·C[a,b] with LL norm, i.e., f=esssupt[a,b]|f(t)|f=esssupt[a,b]|f(t)| is a Banach space.
• Lp[a,b]={f:fp<}Lp[a,b]={f:fp<} for p[1,]p[1,] and -a<b-a<b is a Banach space.
• p(N)={ sequences x:xp<}p(N)={ sequences x:xp<} for p[1,]p[1,] is a Banach space.
• Any finite-dimensional normed vector space is Banach, e.g., RNRN or CNCN with any norm.
• C[a,b]C[a,b] with LpLp norm for p<p< is not Banach.

Definition 2

A complete inner product space is called a Hilbert space.

Example 2

• L2[a,b]L2[a,b] is a Hilbert space.
• 2(N)2(N) is a Hilbert space.
• Any finite-dimensional inner product space is a Hilbert space.

Note that every Hilbert space is Banach, but the converse is not true. Hilbert spaces will be extremely important in this course.

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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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