Let (, || ⋅ ||) be a norm linear space over ℝ. Show that (, )⊊ [, ]
.
Answers
Answer:
ok
Explanation:
The field F of scalars will always be C or R.
2. Definition: A linear space over the field F of scalars is a set V satisfying
a. V is closed under vector addition: For u and v in V , u + v is in V also.
b. Vector addition is commutative and associative: For all u, v and w in V ,
u + v = v + u,
(u + v) + w = u + (v + w).
c. There is a zero element (denoted 0) in V , such that v + 0 = v for all v in V .
d. For each v in V , there an additive inverse −v such that v + (−v) = 0. (Note: We
usually write u − v instead of u + (−v).)
e. V is closed under scalar multiplication: For α ∈ F and u ∈ V , αu ∈ V .
f. Scalar multiplication is associative and distributive: For all α and β in F and u and w
in V ,
α(βu) = (αβ)u,
(α + β)u = αu + βu,
α(u + w) = αu + αw.
g. 1 v = v for all v in V .
3. Example: Rn, with the usual operations, is a vector space over R.
4. Example: Cn, with the usual operations, is a vector space over C.
5. Note: Instead of the previous two examples, we could have simply stated that F
n,
with the usual operations, is a vector space over F.
6. Example: The set C[a, b] of F-valued continuous functions defined on [a, b] is a linear
space over F. (Note: Elements of C[a, b] are continuous from the right at a and from
the left at b.)
7. Example: The set C
k
[a, b] of F-valued k-times continuously differentiable functions
defined on [a, b], is a linear space over F. (Again, the derivatives are taken from the
right at a and from the left at b.)