Normed spaces

Question 1

What is a normed space

Answer 1

Let X be a vector space over F. A norm on X is a function ||.||:X->R such that for all x,y in R and alpha in F,

i ) ||x|| >= 0
ii) ||x|| = 0 if and only if x = 0
iii) ||a x|| = |a| ||x||
||x+y|| =< ||x|| + ||y||

Question 2

What is a unit vector

Answer 2

||x|| = 1

Question 3

What is the standard norm on F^n

Answer 3

Euclidean norm
||(x1, .. ,xn)|| = (sum|xj|^2)^(1/2)

Question 4

What is the standard norm on the vector space of continuous functions

Answer 4

||f|| = sup{|f(x)|: x in M}

Question 5

Standard norm on L^p

Answer 5

||f||_p = (integral_X|f|^p dmu)^(1/p)

Question 6

Standard norm on L^infinity

Answer 6

||f||_infinity =ess sup{|f(x)|:x in X}

Question 7

standard norm on l^p

Answer 7

||{x_n}||_p = (sum|x_n|^p)^(1/p)

Question 8

Standard norm on l^infinity

Answer 8

||{x_n}||_infinity = sup{|x_n|:n in N}

Question 9

Give the definition of equivalent norms

Answer 9

Two norms are equivalent if there exists M,m>0 such that for all x in X

m||x||_1 <= ||x||_2 <= M||x||_1

Question 10

Are 2 norms on a finite dimensional space X equivalent

Answer 10

If ||.Z|| and ||.||_2 are any two norms on a finite-dimensional vector space X then they are equivalent.

Question 11

Does a finite dimensional space having a norm mean the space is complete

Answer 11

Yes

Question 12

What can be said about Y if Y is a finite-dimensional subspace of a normed vector space X,

Answer 12

If Y is a finite-dimensional subspace of a normed vector space X, then Y is
closed.

Question 13

What can be said if X is a normed vector space and S is a linear subspace of X

Answer 13

If X is a normed vector space and S is a linear subspace of X then S is a linear subspace of X.

Question 14

What is the closed linear span of a non-empty subset of X

Answer 14

Let X be a normed vector space and let E be any non-empty subset of X. The closed linear span of E, denoted by \bar{Sp}(E), is the intersection of all the closed linear subspaces of X which contain E.

Question 15

What is Riesz’ Lemma

Answer 15

Suppose that X is a normed vector space, Y is a closed linear subspace of X such that Y \ neq X and α is a real number such that 0 < α < 1. Then there exists x_α ∈ X such that ||x _α|| = 1 and ||x_α − y|| > α for all y ∈ Y

Question 16

If X is a finite dimensional normed space are, D = {x ∈ X : ||x||≤ 1} nor K = {x ∈ X : ||x|| = 1} compact

Answer 16

No

Question 17

Define a Banach space

Answer 17

A Banach space is a normed vector space which is complete under the metric associated with the norm.

Question 18

Any finite-dimensional normed vector space is a Banach space?

Answer 18

Yes

Question 19

Any finite-dimensional normed vector space is a Banach space.

Question 20

Any finite-dimensional normed vector space is a Banach space.

Question 21

is l^p Banach?

Answer 21

Yes

Question 22

If X is a Banach space and Y is a linear subspace of X when is Y Banach

Answer 22

if and only if Y is closed in X

Question 23

Define convergence in normed spaces

Answer 23

Let X be a normed space and let {x_k} be a sequence in X. For each positive integer n let s_n = sum{x_k} from 1 to n be the nth partial sum of the sequence.

We say the series converges if lim_(n -> infinity) s_n exists in X and if so they are defined as ewual

Question 24

What can BE said about a series of norms in a banach space

Answer 24

If the series sum_(k=1)^(infinity)(||x_k||) converges then the sum sum_(k=1)^(infinity)(x_k) also converges