Recommended Functional Analysis Book (Graduate Level)

Functional Analysis is a subject that combines Linear Algebra with Analysis. I researched online, and it seems one of the best Functional Analysis Book for Graduate level is Functional Analysis by Peter Lax. This book is ideal for a second course in Functional Analysis. For a first course in Functional Analysis, I would recommend Kreyszig, which is listed on my Recommended Undergraduate Math Books page.

This is Theorem 5 in the book: Let X be a linear space over the reals.

The following 9 properties hold:

  1. The empty set is convex.
  2. A subset consisting of a single point is convex.
  3. Every linear subspace of X is convex.
  4. The sum of two convex subsets is convex.
  5. If K is convex, so is -K.
  6. The intersection of an arbitrary collection of convex sets is convex.
  7. Let \{K_j\} be a collection of convex subsets that is totally ordered by inclusion. Then their union \cup K_j is convex.
  8. The image of a convex set under a linear map is convex.
  9. The inverse image of a convex set under a linear map is convex.

Brief sketch of proofs:

We give a brief sketch of the idea behind the proofs.

We are using the definition of convex as follows: X is a linear space over the reals; a subset K of X is called convex if, whenever x and y belong to K, all points of the form ax+(1-a)y, 0\leq a\leq 1 also belong to K.

Property 1 is vacuously true.

Property 2 is true because of ax+(1-a)x\equiv x.

Property 3 is true because ax+(1-a)y is a linear combination and is thus in the linear subspace.

Property 4) Let C_1 and C_2 be the two convex subsets. Let x_1+y_1 and x_2+y_2 be points in C_1+C_2=\{x+y:x\in C_1, y\in C_2\}

\begin{aligned}    a(x_1+y_1)+(1-a)(x_2+y_2)&=ax_1+ay_1+x_2+y_2-ax_2-ay_2\\    &=[ax_1+(1-a)x_2]+[ay_1+(1-a)y_2]\\    &\in C_1+C_2    \end{aligned}

Property 5) We just need to know that -K=\{-x:x\in K\} and this algebraic observation: a(-x)+(1-a)(-y)=-(ax+(1-a)y).

Property 6) Let x,y\in\bigcap_{i\in I}C_i. ax+(1-a)y\in C_i for all i\in I, thus ax+(1-a)y\in\bigcap_{i\in I}C_i.

Property 7) Let x,y\in\bigcup K_j, where K_i\subseteq K_{i+1}. Let x\in K_n, y\in K_m, then either K_n\subseteq K_m or K_m\subseteq K_n. If K_n\subseteq K_m, ax+(1-a)y\subseteq K_m\subseteq \bigcup K_j. Similarly for the other case K_m\subseteq K_n.

Property 8) Observe that af(x)+(1-a)f(y)=f(ax+(1-a)y)\in f(K).

Property 9) The only tricky thing about this part is that we cannot assume that the inverse f^{-1} exists. We can only talk about the pre-image.

Let w,z\in f^{-1}(K). f(w)\in K and f(z)\in K.

We have f(aw+(1-a)z)=af(w)+(1-a)f(z)\in K.

Thus aw+(1-a)z\in f^{-1}(K).

The End!

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